2 Elliptic Dierential Equations
2.1 Classical solutions
As far as existence and uniqueness results for classical solutions are con-cerned, we restrict ourselves to linear elliptic second order elliptic dierential equations in a bounded domain Ω ⊂ Rd
(2.1.1) Lu := − d ∑ i,j=1 aij uxixj+ d ∑ i=1 bi uxi+ cu = f .
We assume that the data of the problem satis es the following conditions: There holds aij, bi, c∈ C(Ω), 1 ≤ i, j ≤ d, f ∈ C(Ω) and
(2.1.2a)
there exists a constant C > 0, such that for all x ∈ Ω
|aij(x)| , |bi(x)| , |c(x)| ≤ C , 1 ≤ i, j ≤ d , The functions aij are symmetric, i.e., for all 1 ≤ i, j ≤ d
(2.1.2b)
and x ∈ Ω we have
aij(x) = aji(x) ,
and the matrix-valued function (aij)di,j=1 is uniformly
positive de nite in Ω , i.e., there exists a constant α > 0 , such that for all x ∈ Ω und ξ ∈ Rd
d
∑
i,j=1
aij(x)ξiξj ≥ α |ξ|2 .
For the sake of uniqueness of a solution of (2.1.1) we need to specify boundary conditions on the boundary Γ = ∂Ω.
(2.1.3) De nition: Under the assumptions (2.1.2a),(2.1.2b) let L be the lin-ear second order elliptic dierential operator given by (2.1.1), and assume
f ∈ C(Ω) and uD∈ C(Γ ). Then, the boundary value problem Lu(x) = f (x) , x∈ Ω ,
(2.1.4a)
u(x) = uD(x) , x∈ Γ
is called an inhomogeneous Dirichlet problem. If uD≡ 0, (2.1.4a),(2.1.4b) is
said to be a homogeneous Dirichlet problem. A function u ∈ C2(Ω)∩ C(Ω) is called a classical solution, if u satis es (2.1.4a),(2.1.4b) pointwise.
In case c(x) ≥ 0 , x ∈ Ω, in (2.1.1), the uniqueness of a classical solution of the Dirichlet problem (2.1.4a),(2.1.4b) follows from the Hopf maximum principle.
(2.1.5) Theorem: In addition to the conditions (2.1.2a),(2.1.2b) assume
c(x)≥ 0 , x ∈ Ω. Further, assume that the function u ∈ C2(Ω)satis es
(2.1.6) Lu(x) ≤ 0 , x ∈ Ω .
Then, if there exists x0∈ Ω such that
(2.1.7) u(x0) = sup
x∈Ω
u(x) ≥ 0 ,
there holds
(2.1.8) u(x) = const. , x ∈ Ω .
Proof. We refer to Theorem 2.1.2 in Jost (2002). Corollary:Under the assumptions of Theorem (2.1.5) the Dirichlet problem (2.1.4a),(2.1.4b) admits at most one classical solution.
Proof. The proof is left as an exercise.
For the prototype of a linear second order elliptic dierential equation, the Poisson equation (??), we can prove the existence of a solution of the the Dirichlet problem (2.1.4a),(2.1.4b) using Green's representation formula. We assume Ω to be a C1domain (see De nition (??)).
Green's rst and second formula are as follows:
(2.1.9) Theorem: Let Ω ⊂ Rd be a C1 domain and assume u, v ∈ C2(Ω). Further, let ν be the exterior unit normal vector on Γ . Then, there hold the rst Green's formula (2.1.10) ∫ Ω ∆u v dx + ∫ Ω ∇u · ∇v dx = ∫ Γ ν· ∇u v dσ
and the second Green's formula (2.1.11) ∫ Ω ( ∆u v− u ∆v ) dx = ∫ Γ ( ν· ∇u v − u ν · ∇v ) dσ .
2.1 Classical solutions 3
Proof. The proof of (2.1.10) follows from Gauss' theorem ∫ Ω ∇ · q dx = ∫ Γ ν· q dσ
applied to the vector-valued function q := v∇u. Exchanging the roles of u and v in (2.1.10) and subtracting the resulting equation from (2.1.10) implies
(2.1.11).
(2.1.12) De nition: A function u ∈ C2(Ω) is called a harmonic function in Ω, if u satis es the Laplace equation, i.e., if ∆u(x) = 0 , x ∈ Ω. The function (2.1.13) Γ (x, y) := Γ (|x − y|) := { 1 2π ln(|x − y|) , d = 2 1 d(2−d)|Bd 1| |x − y| 2−d, d > 2 , where x, y ∈ Rd , x̸= y, and |Bd
1| denotes the volume of the unit ball in Rd, is called the fundamental solution of the Laplace equation. As a function of
x, the fundamental solution is a harmonic function in Rd\ {y}.
The notion 'fundamental solution' is justi ed by Green's representation formula:
(2.1.14) Theorem: Let Ω ⊂ Rd be a C1 domain and Γ (·, ·) the fundamental solution as given by (2.1.13). If u ∈ C2(Ω), in y ∈ Ω there holds
u(y) = ∫ Ω Γ (x, y) ∆u(x) dx + (2.1.15) + ∫ ∂Ω ( u(x) ν· ∇xΓ (x, y)− Γ (x, y) ν · ∇u(x) ) dσ(x) ,
where ∇x denotes the gradient with respect to the dierentiation in the
vari-able x.
Proof. The proof follows from Green's second formula (2.1.11). For details
we refer to Theorem 1.1.1 in Jost (2002).
The application of (2.1.15) to a test function φ ∈ C∞
0 (Ω)results in
φ(y) =
∫
Ω
Γ (x, y) ∆φ(x) dx , y∈ Ω .
If we denote by ∆xthe Laplace operator with respect to the variable x and
by δy Dirac's delta function as a distribution with δy(φ) = φ(y), we can
interpret ∆xΓ (·, y) according to
(2.1.16) De nition: A function G(x, y), x, y ∈ Ω , x ̸= y, is called Green's function for Ω, if the following conditions are satis ed
G(x, y) = 0 , x∈ ∂Ω ,
(2.1.17a)
The function G(x, y) − Γ (x, y) is harmonic in x ∈ Ω . (2.1.17b)
For the domains Ω ⊂ Rd considered here, the existence of a Green's
function is guaranteed. Under certain assumptions, it enables the explicit representation of the solution of the inhomogeneous Dirichlet problem for Poisson's equation (??).
(2.1.18) Theorem: Under the assumptions of Theorem (2.1.14) let G(x, y) be a Green's function for Ω. Then, the following holds true:
(i) If u ∈ C2(Ω)∩C(Ω) is a classical solution of the inhomogeneous Dirichlet problem (2.1.4a),(2.1.4b) for Poisson's equation (??), for y ∈ Ω we have the representation (2.1.19) u(y) = ∫ Ω G(x, y) f (x) dx + ∫ ∂Ω uD(x) ν· ∇xG(x, y) dσ(x) .
(ii) If f is a Hölder continuous function in Ω and uD ∈ C(∂Ω), then
the function u given by (2.1.19) is a classical solution of the inhomogeneous Dirichlet problem (2.1.4a),(2.1.4b) for Poisson's equation (??).
Proof. Assertion (i) follows readily from the application of Green's second formula (2.1.11) to the function v(x) = G(x, y) − Γ (x, y). The proof of (ii) requires additional results concerning the regularity of solutions of elliptic dierential equations. We refer to chapters 9.1 and 10.1 in Jost (2002).
2.2 Finite dierence methods
We consider the boundary value problem (2.1.4a),(2.1.4b) under the assump-tions (2.1.2a),(2.1.2b) and c(x) > 0, x ∈ Ω. For the numerical solution we use nite dierence methods on the basis of an approximation of the partial derivatives in (2.1.4a) by dierence quotients (cf. Chapter 3.5 in Stoer, Bu-lirsch (2002)). We begin by assuming that the domain Ω is a d-dimensional cube.
(2.2.1) De nition: Let Ω := (a, b)d, a, b∈ R, a < b, h := (b−a)/(N +1), N ∈
N. Then, the set
Ωh:={(xi1,· · · , xid)
T | xi
j = a + ijh , 0≤ ij ≤ N + 1 , 1 ≤ j ≤ d}
is called a grid-point set with step size h. It represents a uniform or equidistant grid. A grid is called uniform or equidistant, if the grid-points have the same distance from each other. We further refer to Ωh := Ωh∩ Ω as the set of
2.2 Finite dierence methods 5
For the approximation of the rst and second partial derivatives uxi, uxixi
we consider the following dierence quotients:
(2.2.2) De nition: For u : Ω → R and i ∈ {1, · · · , d}, the dierence quo-tients
Dh,i+ u(x) := h−1(u(x + hei)− u(x)) ,
(2.2.3a)
Dh,i− u(x) := h−1(u(x)− u(x − hei)) , (2.2.3b)
Dh,i1 u(x) := (2h)−1(u(x + hei)− u(x − hei)) ,
(2.2.3c)
where ei is the i-th unit vector in Rd, are called the forward, backward and
central dierence quotients for the rst partial derivatives uxi, 1 ≤ i ≤ d,.
The dierence quotients given by
(2.2.4) D2h,i,iu(x) := h−2(u(x + hei)− 2u(x) + u(xi− hei))
are called central dierence quotients for the second derivatives uxixi.
For u ∈ C2(Ω), Taylor expansion shows that the dierence quotients
D±h,iuprovide an approximation of uxi of order O(h), whereas for u ∈ C
4(Ω) the central dierence quotients D1
h,iuand D
2
h,iuresult in an approximation
of uxi and uxixi of order O(h
2).
As far as the approximation of the mixed second partial derivatives uxixj, 1≤
i ̸= j ≤ d, in x ∈ Ωh is concerned, it is obvious to use a weighted sum of
functions values of u in points of the (xi, xj)-plane which are neighbors of x.
This leads to
(2.2.5) D2h,i,ju(x) = ∑ |α|,|β|≤1
κα,βu(x + αhei+ βhej) ,
where κα,β ∈ R , α, β ∈ Z. Assuming su cient smoothness of u and
re-quiring that these dierence quotients approximate the mixed second partial derivatives of order O(h2), by Taylor expansion we obtain the conditions
κ0,0=−(4h2)−1γ1 ,
(2.2.6)
κα,β= (2h2)−1(γ1− sgn(α)γ2− sgn(β)γ3) , |α| + |β| = 1 ,
κα,β= (4h2)−1(sgn(αβ) − γ1+sgn(α)γ2+sgn(β)γ3) , |α| = |β| = 1 ,
where γi ∈ R, 1 ≤ i ≤ 3, are free parameters. Hence, there exist several
possibilities for the approximation of the mixed second partial derivatives of order O(h2). We will discuss later which is the most suited one with regard to the nite dierence approximation of (2.1.4a)-(2.1.4b).
An appropriate tool to characterize dierence quotients are dierence stars. (see Fig. 1). Here, the points displayed by • mark those points used in the dierence quotient and the number next to a point refers to the corresponding weight.
κ−1,1 κ−1,0 κ−1,−1 κ0,1 κ1,1 κ0,0 κ 1,0 κ0,−1 κ1,−1
Fig. 1. Dierence star for mixed second partial derivatives
We also use the following characterization
(2.2.7) κκ−1,1−1,0 κκ0,10,0 κκ1,11,0 κ−1,−1 κ0,−1κ1,−1 .
The dierence quotient given by (2.2.5) is also an example for a so-called compact dierence approximation. A dierence approximation
∑
|α|≤m
καu(x + αh) , α∈ Zd , m∈ N ,
and the associated dierence star are called compact, if besides the grid point x only its next neighbors are used in the dierence approximation, i.e., if m = 1.
An important special case of (2.1.4a) is Poisson's equation (cf. (??)), i.e.,
A = −∆ where ∆ is the Laplace operator ∆ = ∑di=1uxixi. In view of the
de nition of the central dierence quotients for the second partial derivatives
uxixi, we use the following dierence approximation of the Laplace operator:
(2.2.8) De nition: Let D2
h,ibe given as in De nition (??). Then, the
dier-ence operator ∆h := d ∑ i=1 D2h,i
is called the discrete Laplace operator. In case u ∈ C4(Ω), ∆huprovides an approximation of ∆u of order O(h2).
For d = 2, the dierence star associated with the dierence operator −∆h is
displayed in Fig. 2. Since exactly ve grid points are involved in the dierence approximation, the dierence star is referred to as the ve-point dierence star.
In order to derive compact dierence approximations of −∆u in two space dimensions, as in (2.2.5) we consider
2.2 Finite dierence methods 7 −1 h2 4 h2 −1 h2 −1 h2 −1 h2 − 1 3h2 − 1 3h2 − 1 3h2 − 1 3h2 − 1 3h2 8 3h2 − 1 3h2 − 1 3h2 − 1 3h2
Fig. 2. Five-point and nine-point dierence stars for the approximation of the dierential operator −∆ in two space dimensions
∑
|α|,|β|≤1
κα,βu(x + αhei+ βhej) , α, β∈ Z
and try to determine the weights κα,βsuch that we obtain a dierence
appro-ximation of order O(h2)for su ciently smooth u. This leads to the conditions
κ1,1 = κ1,−1 = κ−1,1 = κ−1,−1 ,
κ1,0 = κ0,1 = κ0,−1 = κ−1,0, 4κ1,1 + 4κ1,0 + κ0,0 = 0 , 2κ1,1 + κ1,0 = −1/h2 .
For κ1,1 = 0 and κ1,0 = −1/h2 we get the ve-point dierence star. For
κ1,1 = κ1,0 = −1/(3h2) we obtain the nine-point dierence star displayed in Fig. 2 (right). However, it is not possible to obtain the order O(h3). On the other hand, using non-compact dierence approximations one can realize approximations of higher order than 2.
In case of a more general bounded domain Ω ⊂ Rd with boundary Γ , the
de nition of the set of interior grid points has to be renewed.
(2.2.9) De nition: Let Ω ⊂ Rd be a bounded, simply connected domain with
boundary Γ and Rd
h, h > 0the grid-point set given by
Rd
h := {x = (x1,· · · , xd)T | xi= ih , i∈ Z} .
Then, the set Ωh:=Rdh∩Ω is called the set of interior grid points. A grid point x∈ Γ is called a boundary grid point, if there exists an interior grid point x∗∈ Ωh such that x = x∗+ αhei for some i ∈ {1, · · · , d} and α ∈ R, |α| < 1.
In this case, x∗is said to be next to the boundary. All interior grid points that
are not next to the boundary are called far from the boundary. We denote by
Γh, ΩΓh and Ω o
h the set of boundary grid points, the set of grid points next to
the boundary, and the set of grid points far from the boundary, respectively (see Fig. 3). We set Ωh:= Ωh∪ Γh.
The line between two neighboring grid points along a grid-line is called an edge, and the two grid points are called the vertices of the edge. We call a grid-point set discretely connected, if any two grid points in Ωh can be
connected by a path consisting of a nite number of edges with vertices in
Ωh. Ω xΓ ν x∗Γ x1 x2
Fig. 3. Grid point next to the boundary • and boundary grid points ◦ in a do-main with curved boundary (left) and approximation of the normal derivative in a boundary grid point (right)
Remark.Observe that an interior grid point x can be next to the boundary although x ± hei∈ Ωh, 1≤ i ≤ d.
The de nition of dierence quotients and of the discrete Laplace operator also has to be modi ed. This leads to the Shortley-Weller approximation. (2.2.10) De nition: Let Ωh⊂ Rd be a grid-point set as in De nition (2.2.9)
and x ∈ Ωh, x± h±i ei ∈ Ωh, hi± = α±i h,|α±i | ≤ 1, 1 ≤ i ≤ d. For the
approximation of uxi in x, the forward and backward dierence quotients
are given by Dh,i+ u(x) := ( u(x + h+i ei)− u(x) ) /h+i , Dh,i− u(x) := ( u(x)− u(x − h−i ei) ) /h−i .
The central dierence quotient for the second derivatives uxixi is given by
D2h,iu(x) := 2 h+i + h−i (u(x + h+ i ei)− u(x) h+i + u(x− h−i ei)− u(x) h−i ) .
For the mixed second derivatives uxixj, 1≤ i ̸= j ≤ d, we may proceed as in
(2.2.5).
We de ne the discrete Laplace operator ∆has in De nition (2.2.8). For d = 2
and x = (x1, x2)∈ Ωh such that (x1± α±1h, x2± α±2h)∈ Ωh, we obtain the explicit representation
2.2 Finite dierence methods 9 ∆hu(x) := 2 h2 ( 1 α+1(α + 1 + α−1) u(x1+ α+1h, x2) + 1 α−1(α + 1 + α−1) u(x1− α−1h, x2) + + 1 α+2(α + 2 + α−2) u(x1, x2+ α+2h, ) + 1 α−2(α + 1 + α−2) u(x1, x2− α−2h)− −( 1 α+1α−1 + 1 α+2α−2 )u(x1, x2) ) .
Remark.(i) In grid points next to the boundary, the central dierence quo-tient for the second derivatives and the discrete Laplace operator in general only provide an approximation of uxixi and ∆u of order O(h).
(ii) The de nition of the dierence quotients and the discrete Laplace opera-tor in (2.2.10) can be easily generalized to non-equidistant grids.
For the discretization of the normal derivative uν = ν· ∇u in xΓ ∈ Γh,
where ν denotes the exterior normal unit vector in xν, let x∗Γ ∈ Ωh be the
rst intersection in Ωh of the normal through xΓ and a grid line (cf. Fig. 3
(right)). Let xi, 1≤ i ≤ 2, be those grid points on that grid line next to x∗Γ
assuming that the grid is su ciently ne such that xi ∈ Ωh, 1≤ i ≤ 2. We
refer to ˜u(x∗
Γ) as the value obtained by linear interpolation with respect to u(x1)and u(x2). Then, an approximation of the normal derivative ν · ∇u in
xΓ is given by
(2.2.11) ∂νhu(xΓ) := u(xΓ)− ˜u(x
∗ Γ) |xΓ − x∗
Γ| .
This approach can also be used for directional derivatives uµ = µ· ∇u with
respect to a non-tangential vector µ in xΓ.
We compute approximations within the linear space of functions de ned on a grid-point set.
(2.2.12) De nition: Let Ωh be a grid-point set as in (2.2.10). A function uh: Ωh→ R is called a grid function We refer to C(Ωh)as the linear space
of grid functions. The space C(Ωh) will be equipped with the norm
(2.2.13) ∥uh∥h := ∥uh∥ℓ∞(Ωh) = max
x∈Ωh
|uh(x)| .
We rst consider a nite dierence method for the approximation of Poisson's equation with inhomogeneous Dirichlet boundary conditions by means of the discrete Laplace operator. Given grid functions fh ∈ C(Ωh)
and uD
h ∈ C(Γh), we are looking for uh∈ C(Ωh)such that −∆huh = fh in Ωh ,
(2.2.14a)
uh = uDh auf Γh .
(2.2.14b)
The existence and uniqueness of a solution of (2.2.14a),(2.2.14b) can be shown either using a discrete maximum principle, a discrete Green's function (cf.,
e.g., Jost (2002) or by means of the investigation of the linear algebraic system (2.2.14a),(2.2.14b). Here, we will follow the latter approach. For simplicity, we will restrict ourselves to the case d = 2 and Ω = (a, b)2 and assume a uniform grid Ωh. The unknown values of the grid function uh in the interior
grid points x ∈ Ωh can be interpreted as the components of a vector. The
structure of the resulting linear algebraic system depends on the ordering of the grid points.
(2.2.15) De nition: Assume Ω = (a, b)2, a, b ∈ R, a < b, and let Ωh be a uniform grid of step size h := (b − a)/N, N ∈ N, with interior grid points
x = (xi, xj)T, 1≤ i, j ≤ N, where xi= a + ih. An ordering of the grid points
from 'bottom' to 'top' and from 'left' to 'right' according to
x(i−1)N+j = (a + ih, a + jh)T , 1≤ i, j ≤ N ,
is called a lexicographic ordering. If we rst count the grid points with even
i + j and then those with odd i + j, the ordering is said to be a checkerboard
ordering.
If we order the interior grid points according to De nition (2.2.15), the values of the grid function in the interior grid points can be assigned a vector with nh := N2 components. For simplifying the notation, this vector will
be denoted in the same way as the grid function, i.e., we de ne uh ∈ Rnh
according to uh,i:= uh(xi), 1≤ i ≤ nh. Then, (2.2.14a),(2.2.14b) corresponds
to a linear algebraic system
(2.2.16) Ahuh = bh
with the coe cient matrix Ah ∈ Rnh×nh and a vector bh ∈ Rnh whose
components are given by the function values of fh and uDh.
In case of a lexicographic ordering of the grid points, Ahis the block
tridia-gonal matrix
(2.2.17) Ah = h−2 tridiag(−I, T, −I) ,
where the nhdiagonal blocks T ∈ RN×N are tridiagonal matrices of the form
(2.2.18) T = tridiag(−1, 4, −1)
and the o-diagonal blocks represent the negative N × N unit matrix. In view of (2.2.17) and (2.2.18) it is easily seen that Ahis a symmetric positive
de nite M-matrix. with the eigenvalues (2.2.19) λij(Ah) = 4h−2
(
sin2(iπh/2) +sin2(jπh/2)), 1≤ i, j ≤ N . The components of the associated orthonormal eigenvectors x(ij)∈ Rnh, 1≤
2.2 Finite dierence methods 11
x(ij)kℓ = h
2 sin(ihkπ) sin(jhℓπ) , 1 ≤ k, ℓ ≤ N .
The structure of the linear algebraic system (2.2.17) in case of a checkerboard ordering of the grid points is left as an exercise.
The M-Matrix property of Ah implies the unique solvability of the linear
system (2.2.16) and hence of the nite dierence method (2.2.14a),(2.2.14b). The properties of Ah further imply the convergence of uh to the solution u of Poisson's equation provided fh → f and uDh → u
D for h → 0. We
want to address the issue of the convergence of nite dierence methods in a more general framework. For that purpose we consider such a method for the approximate solution of the boundary value problem (2.1.4a),(2.1.4b) in
Ω = (a, b)2 with respect to a uniform grid Ω
h with step size h > 0. For
the approximation of the second partial derivatives uxixi and the rst partial
derivatives uxi we use central dierence quotients. The mixed second partial
derivatives uxixj, i̸= j, are approximated by dierence quotients of the form
(2.2.5),(2.2.6). This leads to the nite dierence method
Lhuh = fh in Ωh,
(2.2.20a)
uh = uDh on Γh .
(2.2.20b)
Here, Lh stands for the dierence operator
Lhuh(x) :=− 2 ∑ i,j=1 aij(x)Dh,i,j2 uh(x) + 2 ∑ i=1 bi(x)D1h,iuh(x) + c(x)uh(x)
and fh∈ C(Ωh), uDh ∈ C(Γh)are given grid functions.
Assuming the existence and uniqueness of a solution uh, we de ne the
conver-gence and the order of converconver-gence of the nite dierence method as follows: (2.2.21) De nition: Given Ω := (a, b)2, a, b∈ R, a < b, and a uniform grid
Ωh with step size h > 0, let u : Ω → R be the classical solution of the
boundary value problem (2.1.4a),(2.1.4b) and let uh∈ C(Ωh)be the solution
of the nite dierence method (2.2.20a),(2.2.20b). Then, the nite dierence method is said to be convergent, if
∥u − uh∥h = max
x∈Ωh
|u(x) − uh(x)| → 0 for h → 0 .
It is said to be convergent of order p, if for su ciently small h there holds
∥u − uh∥h = max
x∈Ωh
|u(x) − uh(x)| ≤ C hp ,
where C > 0 is a constant independent of h.
Su cient conditions for the convergence can be stated in terms of the con-sistency with the boundary value problem (2.1.4a),(2.1.4b) and the stability
of the nite dierence method. The consistency is measured by means of the local discretization error which is obtained by evaluating the nite dierence method for the solution of the boundary value problem.
(2.2.22) De nition: Let u be the classical solution of the boundary value problem (2.1.4a),(2.1.4b) and let Ah and fh, uDh be the dierence
opera-tor and the grid functions from (2.2.20a),(2.2.20b). Then, the grid function
τh∈ C(Ωh)given by τh(x) := { Lhu(x)− fh(x) , x∈ Ωh u(x)− uD h(x) , x∈ Γh
is called the local discretization error. The nite dierence method (2.2.20a),(2.2.20b) is said to be consistent with the boundary value problem (??),(??), if
max
x∈Ωh
|τh(x)| → 0 as h → 0 .
It is called consistent of order p, if there exists C > 0, independent of h such that
max
x∈Ωh
|τh(x)| ≤ C hp .
(2.2.23) Lemma: Assume that the boundary value problem (2.1.4a),(2.1.4b) has a classical solution u such that u ∈ C4(Ω), Further, suppose that there holds max x∈Ωh |f(x) − fh(x)| = O(h2) , max x∈Γh |uD(x)− uD h(x)| = O(h 2) . Then, the nite dierence method (2.2.20a),(2.2.20b) is consistent with the boundary value problem (2.1.4a),(2.1.4b) of order p = 2.
Proof. Under the assumptions on fh and uDh the proof follows easily
taking into account the properties of the dierence quotients used in the
de nition of the dierence operator Lh.
Remark.The result of Lemma (2.2.23) can be generalized without di cul-ties to higher dimensions. In case of more general bounded domains and the use of the Shortley-Weller approximation, we can only expect the order of consistency p = 1.
For linear nite dierence methods the stability corresponds to the asymptotically uniform boundedness of the inverse of the dierence operator. (2.2.24) De nition: The nite dierence method (2.2.20a),(2.2.20b) is called stable, if there exists hmax > 0 such that for h ≤ hmax the operator Lh is
invertible and there is a constant C > 0 independent of h such that
∥L−1
2.2 Finite dierence methods 13
(2.2.25) Theorem: Assume that (2.2.20a),(2.2.20b) is stable and consistent with the boundary value problem (2.1.4a),(2.1.4b). Then, the nite dierence method is convergent and the order of convergence is the same as the order of consistency.
Proof. The proof is an immediate consequence of
Lh(u(x)− uh(x)) = τh(x) , x∈ Ωh ,
and u(x) − uh(x) = τh(x), x∈ Γh.
If the nite dierence method is consistent with the boundary value pro-blem, a su cient condition for stability is the M-matrix property of the ma-trix associated with the nite dierence method.
(2.2.26) Lemma: Assume that (2.2.20a),(2.2.20b)is consistent with the bound-ary value problem (2.1.4a),(2.1.4b), and let Ah ∈ Rnh×nh be the coe cient
matrix of the linear algebraic system associated with the nite dierence method. If Ah is an M-matrix, then the nite dierence method is stable.
Proof. The M-matrix property implies that for vectors zh ∈ Rnh such
that zh> 0and Ahzh> 0there holds
(2.2.27) ∥A−1 h ∥ ≤ ∥zh∥∞ min 1≤i≤nh (Ahzh)i ,
where ∥ · ∥ stands for the row-sum norm (cf., e.g., Berman and Plemmons (1979).) Now, assume that u∗is the solution of the boundary value problem
(2.1.4a),(2.1.4b) for f = 1 and uD= 0. According to the maximum principle
due to Hopf (??) we have u∗(x) > 0, x ∈ Ω. Moreover, Lu∗(x)≥ 1, x ∈ Ω.
We de ne the vector zh according to zh := (u∗(x1),· · · , u∗(xnh))
T. In view
of the consistency, for su ciently small h it follows that Ahzh≥ 1/2. Hence,
(2.2.27) implies the asymptotically uniform boundedness of A−1
h and thus the
stability of the nite dierence method.
According to Lemma (2.2.26) the stability of the nite dierence method follows from the M-matrix property of Ah∈ Rnh×nh. It is easily shown that
for a12(x) = 0, x∈ Ω, the dierence operator Lh gives rise to the dierence star 1 h2 0 h 2b2− a22 0 −h 2b1− a112(a11+ a22) h 2b1− a11 0 −h2b2− a22 0 .
Assuming h|bi(x)| < 2aii(x), 1 ≤ i ≤ 2, Ah turns out to be an M-matrix.
However, for non-vanishing a12 we have to choose the dierence quotients
D2
h,i,juh(x), 1≤ i ̸= j ≤ 2, from (2.2.5) with (2.2.6) depending on the sign of a12(x). For γ1=±1, γ2= γ3= 0in (2.2.6) we obtain the dierence stars
1 2h2 −1 1 01−2 1 0 1−1 , 1 2h2 −1 2 −10−1 1 1−1 0 . We denote by D2,− h,i,j and by D 2,+
h,i,j the dierence operators associated with
the left and right dierence star. Then, if ±a12(x) > 0and we use the dier-ence quotient D2,±
h,i,juh(x)in the de nition of Ah, it is easily shown that the
condition
(2.2.28) h|bi(x)| < 2 (aii(x)− |a12(x)|) , 1 ≤ i ≤ 2 , x ∈ Ω implies the M-matrix property of Ah.
(2.2.29) Lemma: Let Ah be the dierence operator from (2.2.20a) such that D2
h,i,juh(x) = D
2,±
h,i,juh(x), 1≤ i ̸= j ≤ 2, if ±a12(x) > 0, x∈ Ω, and suppose that (2.2.28) holds true. Then, the nite dierence method (2.2.20a),(2.2.20b) is stable.
Proof. Condition (2.2.28) implies the M-matrix property of the matrix associated with the dierence operator and hence, the assertion follows from
Lemma (2.2.26).
In summary we obtain the following result:
(2.2.30) Theorem: Under the assumptions of Lemma (2.2.23) and Lemma (2.2.29) the nite dierence method (2.2.20a),(2.2.20b) is convergent of order
p = 2.
Proof. Lemma (2.2.23) proves the consistency of order p = 2 and Lemma (2.2.29) guarantees the stability of the nite dierence method. Theorem
(2.2.25) allows to conclude.
Remark.(i) For general bounded domains and the Shortley-Weller approxi-mation the stability of nite dierence methods can be shown under appro-priate conditions on the coe cient functions of the dierence operator. Ac-cording to Remark (10.3.2.26) the order of convergence reduces to p = 1. (ii) Condition (2.2.28) does not hold true, if the coe cient function bi
signi -cantly dominates the coe cient function aiiin absolute value. Such a problem
is called convection-dominated, since for diusion-convection-reaction prob-lems as described by (2.1.4a) the term b · ∇u represents the convective part of the process. For such problems, the discretization of the rst partial deri-vatives is done by the forward or the backward dierence quotient depending on the sign of bi (upwind discretization).
The convergence analysis presented in this section relies on smoothness assumptions with respect to the classical solution of the boundary value pro-blem. For many problems, such assumptions do not hold true, e.g., in case of singularities due to inconsistent boundary conditions or due to the geome-try of the domain. As far as the development, analysis and implementation of nite dierence methods for such problems is concerned, we refer to Samarskii, Lazarov and Makarov (1987).
2.3 Weak solutions 15
2.3 Weak solutions
We consider a boundary value problem for a linear second order elliptic dif-ferential operator in selfadjoint form with homogeneous Dirichlet boundary conditions Lu = f in Ω, (2.3.1a) u = 0 on Γ, (2.3.1b) where Lu :=−∇ ( a∇u ) + cu.
We assume Ω ⊂ Rd to be a bounded Lipschitz domain with boundary Γ = ∂Ω. We further suppose that a = (aij)di,j=1 with aij ∈ L∞(Ω), 1 ≤ i, j ≤ d,
is a uniformly positive de nite matrix-valued function, i.e., there exists a constant α > 0 such that for almost all x ∈ Ω
(2.3.2)
d
∑
i,j=1
aij(x)ξiξj≥ α|ξ|2 , ξ = (ξ1,· · · , ξd)T ∈ Rd.
Moreover, we assume c ∈ L∞(Ω)such that c(x) ≥ 0 for almost all x ∈ Ω and f ∈ L2(Ω).
Multiplication of (2.3.1a) by a function v ∈ C∞
0 (Ω), integration over Ω and the application of Green's formula yields
(2.3.3) ∫ Ω ( a∇u · ∇v + cuv ) dx = ∫ Ω f v dx.
Obviously, the integrals in (2.3.3) are well de ned for functions u and v in the Sobolev space H1
0(Ω)(see section (1.2)). Therefore, we introduce the notion of weak (generalized) solution.
(2.3.4) De nition: A function u ∈ H1
0(Ω) is called a weak or generalized solution of (2.3.1a),(2.3.1b), if (2.3.3) is satis ed for all v ∈ H1
0(Ω). In case f ∈ C(Ω) it is obvious that a classical solution of (2.3.1) in the sense of De nition ?? also represents a solution in the weak sense. On the other hand, if u ∈ H1
0(Ω) is a weak solution such that u ∈ H2(Ω) , the application of Green's formula to (2.3.3) yields
∫
Ω
(
Lu− f)v dx = 0 , v ∈ H01(Ω),
and hence, Lu(x) = f(x) for almost all x ∈ Ω.
(2.3.5) De nition: A function u such that Lu ∈ L2(Ω), which satis es (2.3.1a) for almost all x ∈ Ω, is called a strong solution.
If f ∈ C(Ω) and if a weak solution satis es u ∈ Hm(Ω) for
suf- ciently large m such that Hm(Ω) ,→ C2(Ω)∩ C
0(Ω), it follows that
Lu(x) = f (x), x ∈ Ω, and u(x) = 0, x ∈ Γ , i.e., u is a solution in the
classical sense.
We observe that a weak solution is a solution in the strong or in the classical sense only if it has additional smoothness properties. The study of the regu-larity of weak solutions is a central issue in the theory of partial dierential equations (cf. e.g, Evans (2000)).
The left-hand side in (2.3.3) is a bilinear form on H1
0(Ω)×H01(Ω), whereas the right-hand side represents a bounded linear functional on H1
0(Ω). There-fore, we study the existence and uniqueness of weak solutions within the framework of variational equations in Hilbert spaces:
(2.3.6) De nition: Let V be a real Hilbert space with inner product (·, ·)V
and associated norm ∥ · ∥V, let a(·, ·) : V × V → R be a bilinear form and ℓ∈ V∗. The problem to nd a function u ∈ V such that
(2.3.7) a(u, v) = ℓ(v) , v∈ V,
is called a variational equation.
As we shall see now, the existence and uniqueness of a solution of the vari-ational equation (2.3.7) essentially requires the boundedness and V -ellipticity of the bilinear form.
(2.3.8) De nition: A bilinear form a(·, ·) : V ×V → R is said to be bounded, if there exists a constant C > 0 such that
(2.3.9) |a(u, v)| ≤ C ∥u∥V ∥v∥V , u, v∈ V.
(2.3.10) De nition: A bilinear form a(·, ·) : V × V → R is called V - elliptic, if there exists a constant α > 0 such that
(2.3.11) |a(u, u)| ≥ α ∥u∥2
V , u∈ V.
The following celebrated Lemma of Lax-Milgram provides an existence and uniqueness result for variational equations of the form (2.3.7).
(2.3.12) Theorem: Let V be a Hilbert space with dual V∗ and suppose that a(·, ·) : V × V → R is a bounded, V -elliptic bilinear form and ℓ ∈ V∗. Then,
the variational equation (2.3.7) has a unique solution u ∈ V .
Proof. We denote by A : V → V∗ the operator associated with the
bilinear form according to
2.3 Weak solutions 17
The operator A is a bounded linear operator, since in view of (2.3.9) there holds ∥Au∥V∗ = sup v̸=0 |Au(v)| ∥v∥V ≤ C ∥u∥V, and hence, ∥A∥ = sup u̸=0 ∥Au∥V∗ ∥u∥V ≤ C.
We will reformulate the variational equation (2.3.7) as an operator equation in V by means of the Riesz mapping τ : V∗ → V , ℓ 7−→ τℓ which is given
by
(2.3.14) ℓ(v) = (τ ℓ, v)V , v∈ V.
It can be easily veri ed that the Riesz mapping is an isometry, i.e., (2.3.15) ∥τℓ∥V =∥ℓ∥V∗ , ℓ∈ V∗.
Indeed, using the Riesz mapping, the variational equation (2.3.7) can be equivalently written as the operator equation
(2.3.16) τ (Au− ℓ) = 0 .
We prove the existence and uniqueness of a solution of (2.3.16) by an ap-plication of Banach's xed point theorem. For that purpose we consider the mapping T : V → V de ned by means of
(2.3.17) T v := v − ρ (τ(Av − ℓ)) , ρ > 0.
Obviously, u ∈ V solves (2.3.16) if and only if u ∈ V is a xed point of the operator T . For the proof of the existence and uniqueness of a xed point in
V it su ces to show that the operator T is a contraction on V , i.e., there
exists a constant 0 ≤ q < 1 such that
∥T v1− T v2∥V ≤ q ∥v1− v2∥V , v1, v2∈ V. (2.3.18)
Setting w := v1− v2 and taking (2.3.13) as well as (2.3.15) into account it follows that ∥T w∥2 V =∥w − ρ τ Aw∥ 2 V = =∥w∥2− 2 ρ (τAw, w)V + ρ2 ∥τAw∥2V = =∥w∥2− 2 ρ a(w, w) + ρ2∥Aw∥2V∗ ≤ ≤(1− 2 ρ α + ρ2 C2 ) ∥w∥2 V. We thus obtain
(
1− 2 ρ α + ρ2C2
)
< 1 ⇐⇒ ρ < 2 α C2,
which proves the assertion.
Example. We consider the elliptic boundary value problem (2.3.1). The associated bilinear form a(·, ·) : H1
0(Ω)× H01(Ω)→ R is given by a(u, v) := ∫ Ω ( a∇u · ∇v + cuv ) dx , u, v∈ H01(Ω).
in view of aij ∈ L∞(Ω), 1≤ i, j ≤ d, and c ∈ L∞(Ω) we easily deduce the
boundedness. On the other hand, (2.3.2) and c(x) ≥ 0 f.a.a. x ∈ Ω readily imply
a(u, u)≥ α|u|21,Ω , u∈ H 1 0(Ω). Since on H1
0(Ω)the |·|1,Ω-norm is equivalent to the ∥·∥1,Ω-norm, the previous inequality proves the H1
0(Ω)-ellipticity. Finally, for f ∈ L2(Ω) the linear functional ℓ(v) :=∫Ωf vdx is bounded on H1
0(Ω). The Lax-Milgram lemma (Theorem 2.3.12) implies the existence and uniqueness of a weak solution of (2.3.1).
2.4 Conforming nite element methods
We assume that V ⊆ H1(Ω) is a Hilbert space, a(·, ·) : V × V → R a bounded, V-elliptic bilinear form and ℓ : V → R a bounded linear functional. The Lemma of Lax-Milgram (2.3.12) implies that the variational equation
(2.4.1) a(u, v) = ℓ(v) , v∈ V
admits a unique solution u ∈ V . The Galerkin method consists in the approx-imation of u ∈ V by an approximate solution uh∈ Vhin a nite dimensional
subspace Vh⊂ V, dim Vh= nh, which is the solution of the variational
equa-tion (2.4.1) restricted to Vh, i.e.,
(2.4.2) a(uh, vh) = ℓ(vh) , vh∈ Vh.
Another application of the Lemma of Lax-Milgram reveals the existence and uniqueness of uh∈ Vh.
Remark.If the bilinear form a(·, ·) : V × V → R is symmetric, the Galerkin method is also referred to as the Ritz-Galerkin method.
Specifying a basis (φ(i)
h ) nh
i=1 of Vh, it is easily seen that the Galerkin method
leads to linear algebraic system. Since the solution uh∈ Vh of (2.4.2) can be
2.4 Conforming nite element methods 19 (2.4.3) uh = nh ∑ j=1 αj φ(j)h
uh satis es the variational equation (2.4.2) if and only if
(2.4.4) nh ∑ j=1 a(φ(j)h , φ(i)h ) αj = ℓ(φ (i) h ) , 1≤ i ≤ nh .
Obviously, (2.4.4) represents a linear algebraic system
(2.4.5) Ah αh = bh
in the unknown vector αh= (α1, ..., αnh)
T. The stiness matrix A
h∈ Rnh×nh
and the load vector bh= (bh,1,· · · , bh,nh)
T ∈ Rnh are given by (2.4.6) Ah := a(φ(1)h , φ(1)h ) · · · a(φ(nh) h , φ (1) h ) · · · · · · · · · · a(φ(1)h , φ(nh) h )· · · a(φ (nh) h , φ (nh) h ) and (2.4.7) bh,i := ℓ(φ (i) h ) , 1≤ i ≤ nh.
Remark. We remark that the notions 'stiness matrix' and 'load vector' are stemming from mechanical applications.
With regard to the appropriate choice of nite dimensional subspaces, there are essentially two aspects, namely the
e cient numerical solution of the linear algebraic system (2.4.5) and the accuracy of the approximation of the solution u ∈ V of (2.4.1) by the
solution uh∈ Vh of (2.4.2).
The latter aspect will be addressed in Céa's Lemma. Céa's Lemma states that under the assumptions of the Lax-Milgram Lemma the accuracy of the approximate solution corresponds to the best approximation of the solution
u ∈ V by a function in Vh. Hence, the issue is reduced to a problem in
approximation theory. Céa's Lemma is based on the observation that the error u − uhis a-orthogonal with respect to Vh, i.e.,
(2.4.8) a(u− uh, vh) = 0 , vh∈ Vh .
This property is called the Galerkin orthogonality.
(2.4.9) De nition: If the bilinear form a(·, ·) is symmetric and V-elliptic, it de nes an inner product (·, ·)a := a(·, ·) and an associated norm ∥ · ∥a := a(·, ·)1/2on V which is called the energy norm. Galerkin orthogonality (2.4.8) means that the solution uh ∈ Vh of (2.4.2) is the projection of the solution u∈ V of (2.4.1) onto Vh. Therefore, the approximate solution is referred to
(2.4.10) Theorem: Under the assumptions of the Lemma of Lax-Milgram (2.3.12) let u ∈ V and uh∈ Vhbe the unique solutions of (2.4.1) and (2.4.2).
Then, there holds
(2.4.11) ∥u − uh∥V ≤ C
α vhinf∈Vh
∥u − vh∥V .
Proof. Using the V-ellipticity and the boundedness of a(·, ·) as well as the Galerkin orthogonality (2.4.8), for an arbitrary vh∈ Vhwe have
α∥u − uh∥2V ≤ a(u − uh, u− uh) =
= a(u− uh, u− vh) + a(u− uh, vh− uh)
| {z }
= 0
≤ ≤ C ∥u − uh∥V ∥u − vh∥V ,
which implies (2.4.11).
The e cient numerical solution of the linear algebraic system (2.4.5) re-sulting from the Galerkin method depends on the structure of the stiness matrix Ah. In particular, we are interested in sparsely populated matrices
with a speci c sparsity pattern which can be realized by an appropriate choice of the basis functions of Vh⊂ V with minimal support. In case of conforming
nite element methods, Vhis constructed on the basis of triangulations of the
computational domain Ω ⊂ Rd.
(2.4.12) De nition: Let Ω ⊂ Rd be a bounded Lipschitz domain. A
triangu-lation Th(Ω)of Ω is a partition of Ω into a nite number of subsets T ,called
nite elements such that
Ω = ∪ T∈Th(Ω) T , (2.4.13a) T = T , int(T ) ̸= ∅ , T ∈ Th(Ω) , (2.4.13b)
int(T1) ∩ int(T2) = ∅ for all T1, T2∈ Th(Ω) , T1 ̸= T2 , (2.4.13c)
∂T , T∈ Th(Ω) is Lipschitz continuous . (2.4.13d)
We distinguish between two types of nite elements: simplicial elements and quadrilateral elements.
(2.4.14) De nition: A simplex T in Rd is the convex hull of d + 1 points aj= (aij)di=1∈ Rd: T = { x = d+1 ∑ j=1 λj aj | 0 ≤ λj≤ 1 , d+1 ∑ j=1 λj = 1} .
A simplex T is called non-degenerate, if every point x ∈ Rd can be uniquely
2.4 Conforming nite element methods 21 x = d+1 ∑ j=1 λj aj , λj ∈ R , d+1 ∑ j=1 λj = 1 .
In R2, a simplex is a triangle, in R3 a tetrahedron.
Remark. The non-degeneracy of a simplex is related to the unique solv-ability of the linear algebraic system
a11· · · a1,d+1 · · · · · · · · · · ad1· · · ad,d+1 1 · · · 1 | {z } =: A λ1 · · λd λd+1 = x1 · · xd 1 . (2.4.15)
Obviously, T is non-degenerate if and only if the matrix A is non-singular. (2.4.16) De nition: The points aj, 1≤ j ≤ d + 1, of a simplex T are called
vertices. An m-dimensional face of a simplex T , 0 ≤ m ≤ d, is a simplex in Rm, whose vertices are vertices of T in D. A one-dimensional face is said
to be an edge. For a subset D ⊆ Ω we denote by Fh(D) and Eh(D) the sets
of the (d − 1)-dimensional faces and one-dimensional edges of Th(Ω). The
simplex ˆT with the vertices ˆa1= (0, ..., 0)T and ˆai+1 = ei, 1≤ i ≤ d, is called
simplicial reference element.
(2.4.17) De nition: The barycentric coordinates λj, 1≤ j ≤ d+1, of a point x∈ Rd with respect to the d+1 vertices aj of a non-degenerate simplex T are
the components of the unique solution of the linear algebraic system (2.4.15). The center of gravity xS of a non-degenerate simplex T is the point satisfying
λj(xS) =
1
d + 1 , 1≤ j ≤ d + 1 .
(2.4.18) Lemma: Any non-degenerate simplex T ⊂ Rd is the image of the
simplicial reference element ˆT under an a ne transformation FT : Rd → Rd
ˆ
x7−→ FT(ˆx) = BTx + bˆ T
with a non-singular matrix BT ∈ Rd×dand a vector bT ∈ Rd.
Proof. The proof is left as an exercise.
(2.4.19) De nition: A triangulation Th(Ω) of a polyhedral domain Ω ⊂ Rd
is called a simplicial triangulation, if any of its elements is a non-degenerate simplex in Rd.
An alternative to simplicial triangulations is to use quadrilateral elements. (2.4.20) De nition: A quadrilateral element T in Rd is the tensor product
of d intervals [ci, di], ci≤ di, 1≤ i ≤ d, i.e.,
T = d
∏
i=1
[ci, di] = { x = (x1, ..., xd)T | ci≤ xi≤ di , 1≤ i ≤ d } . A quadrilateral element in R2 is a rectangle, in R3 cube. A quadrilateral element T is said to be non-degenerate, if ci< di, 1≤ i ≤ d.
The quadrilateral element ˆT := [0, 1]d (unit cube) is called the quadrilateral
reference element in Rd. The points aj = (aj1, ..., ajd)
T
, aji = ci or aji= di , 1≤ i ≤ d,
of a quadrilateral element T are called vertices. An m-dimensional face of a quadrilateral element T, 1 ≤ m ≤ d − 1, is a quadrilateral element in Rm
whose vertices are vertices of T . A one-dimensional face is said to be an edge. For D ⊆ Ω we refer to Fh(D) and Eh(D) as the sets of (d − 1)-dimensional
faces and one-dimensional edges of Th(Ω)in D.
(2.4.21) Lemma: Any non-degenerate quadrilateral element T ⊂ Rd is the
image of the quadrilateral reference element ˆT under a diagonal-a ne
trans-formation
FT : Rd → Rd ˆ
x7−→ FT(ˆx) = BTx + bˆ T
with a non-singular diagonal matrix BT = (bTii)di=1 and a vector bT ∈ Rd.
Proof. The proof is left as an exercise.
(2.4.22) De nition: A triangulation Th(Ω) of a quadrilateral domain Ω ⊂
Rdis called a quadrilateral triangulation, if any of its elements T is a
quadri-lateral element.
Given a triangulation Th(Ω), conforming nite element functions will be
speci ed locally for the elements T ∈ Th(Ω) and then composed such that
the resulting global function belongs to the underlying function space V . (2.4.23) De nition: Let Th(Ω)be a triangulation of Ω ⊂ Rdand T ∈ Th(Ω).
Moreover, let PT be a linear space of functions p : T → R such that dim PT = nT. The elements of PT are called local trial functions. For linear bounded
functionals ℓi: PT → R, 1 ≤ i ≤ nT,we de ne
(2.4.24) ΣT := { ℓi(p) | p ∈ PT , 1≤ i ≤ nT } .
2.4 Conforming nite element methods 23
(2.4.25) De nition: Let Th(Ω) be a triangulation of Ω ⊂ Rd, T ∈ Th(Ω)
and PT, ΣT as in De nition (2.4.23). Then, the triple (T, PT, ΣT)is called a
nite element.
A nite element (T, PT, ΣT)is said to be unisolvent, if any p ∈ PT is uniquely
determined by the degrees of freedom in ΣT.
Remark. We remark that the notion ' nite element' is used for T ∈ Th(Ω)
as well as for the triple (T, PT, ΣT).
The numerical computation of the elements of the stiness matrix and the components of the load vector is greatly facilitated in case of a ne equivalent nite elements.
(2.4.26) De nition: Let Th(Ω) be a triangulation of Ω ⊂ Rd, T ∈ Th(Ω)
and PT, ΣT as in De nition (??). Moreover, let ( ˆT , ˆPTˆ, ˆΣTˆ) be a reference element. Then, the nite elements (T, PT, ΣT), T ∈ Th(Ω) are called a ne
equivalent, if there exists an invertible a ne mapping FT : Rd → Rd such
that for all T ∈ Th(Ω) T = FT( ˆT ) ,
(2.4.27a)
PK = {p : T → R | p = ˆp ◦ FT−1 , ˆp∈ ˆPTˆ} ,
(2.4.27b)
ΣT = {ℓi: PT → R | ℓi= ˆℓi◦ FT−1 , ˆℓi∈ ˆΣTˆ , 1≤ i ≤ nT} . (2.4.27c)
(2.4.28) De nition: Let Th(Ω)be a triangulation of Ω ⊂ Rd, T ∈ Th(Ω)and PT, ΣT as in De nition (refdef:10.3.4.25). Then,
(2.4.29) Vh := { vh: Ω→ R | vh|T ∈ PT , T ∈ Th(Ω)}
is said to be a nite element space.
The following result provides su cient conditions for a nite element space Vh to be a subspace of H1(Ω).
(2.4.30) Theorem: Let Vh be a nite element space and assume that there
holds PT ⊂ H1(T ) , T∈ Th(Ω) , (2.4.31a) Vh⊂ C(Ω) . (2.4.31b) Then, we have (2.4.32) Vh⊂ H1(Ω) .
Proof. Let vh ∈ Vh. Obviously, we have vh ∈ L2(Ω). It remains to be
shown that vh has weak rst derivatives whα∈ L
(2.4.33) ∫ Ω vh Dαz dx = (−1)|α| ∫ Ω wαh z dx , z∈ C0∞(Ω) .
Since vh|T ∈ H1(T ), we apply Green's formula elementwise and obtain ∫ Ω vhDαz dx = ∑ T∈Th(Ω) ∫ T vh Dαz dx = (2.4.34) = − ∑ T∈Th(Ω) ∫ T Dαvh z dx + ∑ T∈Th(Ω) ∫ ∂T vh ναz dσ = = − ∑ T∈Th(Ω) ∫ T Dαvh z dx + ∑ F∈Fh(Ω) ∫ F [vh] ναz dσ ,
where [vh] denotes the jump [vh] := vh|T1− vh|T2 across F = T1∩ T2, Ti ∈ Th(Ω), 1≤ i ≤ 2,. But vh∈ C(Ω) and hence, [vh] = 0in (2.4.34) which shows (2.4.33) with wαh|T := Dαvh|T, T ∈ Th(Ω).
Corollary:Let Vh be a nite element space and assume that PT ⊂ H1(T ) , T ∈ Th(Ω) , (2.4.35a)
Vh⊂ C0(Ω) . (2.4.35b)
Then, there holds
(2.4.36) Vh⊂ H01(Ω) .
Proof. The proof is an immediate consequence of Theorem (2.4.30). In the following we will be concerned with the construction of simplicial Lagrangian nite elements. The local trial functions are chosen as polynomials of degree k: Let T be a simplex in Rd. For k ∈ N
0 we de ne Pk(T ) as den
the linear space of polynomials of degree ≤ k in T , i.e., p ∈ Pk(T ), if p(x) = ∑ |α|≤k aαxα , aα∈ R , |α| ≤ k , where xα := d ∏ i=1 xαi i , α := (α1, ..., αd)∈ Nd0 , |α| := d ∑ i=1 αi . We note that (2.4.37) dim Pk(T ) = ( k + d d ) = (k + d)! d! k! .
2.4 Conforming nite element methods 25
(2.4.38) De nition: Let T be a simplex in Rd. Denoting by ai, 1≤ i ≤ d+1,
the vertices of T , the set
(2.4.39) Lk(T ) := { x = d+1 ∑ i=1 λiai| λi∈ {0, 1 k, ..., k− 1 k , 1} , d+1 ∑ i=1 λi= 1}
is called the principal lattice of order k of T . We have card Lk(T ) = ( k + d d ) .
The elements of the principal lattice of order k are called nodal points. For
D⊆ Ω we denote by Nh(D) the set of nodal points in D.
De ning ΣT as the set of degrees of freedom as given by the values of p ∈ Pk(T ) in the nodal points, we obtain a simplicial Lagrangian nite element. (2.4.40) De nition: Let Th(Ω)be a simplicial triangulation of Ω ⊂ Rd. The
triple (T, Pk(T ), Lk(T )), T ∈ Th(Ω), is called a simplicial Lagrangian nite
element of type (k).
(2.4.41) Lemma: A simplicial Lagrangian nite element of type (k) is uni-solvent.
Proof. The proof is left as an exercise. D
(2.4.42) Lemma: Simplicial nite elements of type (k) are a ne-equivalent to the simplicial reference element ( ˆT , ˆPTˆ, ˆΣTˆ).
Proof. Let FT : Rd → Rd be the invertible a ne mapping such that T = FT( ˆT ). Then, there holds Pk(T ) ={p = ˆp◦FT−1| ˆp ∈ Pk( ˆT )}. Since x ∈ Lk(T )
i x = FT(ˆx), ˆx∈ Lk( ˆT ), we also have (2.4.27c).
Remark. In case d = 2, a simplicial Lagrangian nite element of type (1) is called Courant's triangle.
(2.4.43) De nition: The nite element space Vhconsisting of simplicial
La-grangian nite elements of type (k) is called simplicial LaLa-grangian nite ele-ment space and will be denoted by Sk(Ω,Th(Ω)).
In order to guarantee the conformity of Sk(Ω,Th(Ω))we have to assume
that the simplicial triangulation Th(Ω)is geometrically conforming.
(2.4.44) De nition: A simplicial triangulation Th(Ω)is said to be
geometri-cally conforming, if the intersection of two distinct elements of the triangu-lation is either empty or consists of a common face or a common edge or a common vertex.
(2.4.45) Lemma: Let Th(Ω) be a geometrically conforming triangulation.
Then, there holds
(2.4.46) Sk(Ω,Th(Ω)) ={vh∈ C(Ω) | vh|T ∈ Pk(T ), T ∈ Th(Ω)} ⊂ H1(Ω).
Proof. Since Pk(T )⊂ H1(T ) , T ∈ Th(Ω), and in view of Theorem (2.4.30)
we have to show that Sk(Ω,Th(Ω))⊂ C(Ω). The proof will be given in case d = 2 and k = 2: Let Ti ∈ Th(Ω), 1 ≤ i ≤ 2, be two adjacent elements
such that E = T1∩ T2 ∈ Eh(Ω) with nodal points aj ∈ Nh(E), 1≤ j ≤ 3.
Moreover, assume pi∈ P2(Ki), 1≤ i ≤ 2. Then pi|E∈ P2(E). Since p1(aj) = p2(aj), 1≤ j ≤ 3, we conclude that p1|E≡ p2|E. Corollary:We de ne the subspace
Sk,0(Ω,Th(Ω)) :={vh∈ Sk(Ω,Th(Ω))| (2.4.47)
vh|∂T∩∂Ω = 0, T ∈ Th(Ω), T ∩ ∂Ω ̸= ∅} . Under the same assumptions as in Lemma (2.4.45) there holds
(2.4.48) Sk,0(Ω,Th(Ω))⊂ H01(Ω). It remains to specify the nodal basis functions:
(2.4.49) De nition: Let Nh(Ω) = {xj | 1 ≤ j ≤ nh}. The function φi ∈ Sk(Ω,Th(Ω))as given by
(2.4.50) φi(xj) = δij , 1≤ i, j ≤ nh ,
is called the nodal basis function associated with the nodal point xi. The set of
these functions is a basis of the Lagrangian nite element space Sk(Ω,Th(Ω)).
The representation of nodal basis functions by means of barycentric co-ordinates as well as a geometric characterization are left as exercises.
We will now deal with the construction of quadrilateral Lagrangian nite elements: Let T := ∏d
i=1
[ci, di]⊂ Rdbe a quadrilateral element. For k ∈ N0 we denote by Qk(T )the linear space of polynomials of degree ≤ k in each of the dvariables xi, 1≤ i ≤ d, i.e., p ∈ Qk(T ) is of the form
(2.4.51) p(x) = ∑ αi≤k aα1...αd x α1 1 x α2 2 ...x αd d , whence (2.4.52) dim Qk(T ) = (k + 1)d .
2.4 Conforming nite element methods 27
(2.4.53) De nition: Let L[k](T ) denote the point set
L[k](T ) := {x = (xi1, ..., xid) T | xi ℓ := cℓ+ iℓ k(dℓ− cℓ) , (2.4.54) iℓ∈ {0, 1, ..., k} , 1 ≤ ℓ ≤ d}
and let Σ(T ) be given by
(2.4.55) ΣT := {p(x) | x ∈ L[k](T )} , p ∈ Qk(T ) .
The element (T, Qk(T ), ΣT) is called a quadrilateral nite element of type
[k]. and will be denoted by S[k](K). We refer to x ∈ L[k](T )as the associated nodal points.
(2.4.56) Lemma: The Lagrangian nite element of type [k] is based on a tensor product polynomial interpolation of Lagrangian type. The polynomials
p∈ Qk(T ) interpolating in x ∈ L[k](T ) admit the representation
(2.4.57) p(x) = ∑
y∈L[k](T )
L(x) p(y) ,
where L ∈ Qk(T ) stands for the Lagrangian fundamental polynomials
(2.4.58) L(x) =
d
∏
ℓ=1
Liℓ(x) .
Here, Li, 1≤ i ≤ d, are the one-dimensional Lagrangian fundamental
poly-nomials (cf. (2.1.1.3) in Stoer and Bulirsch (2002)).
Proof. Assume p ∈ Qk(T ) and let e ⊂ T be an edge of T according
e := [cj, dj]×
d
∏
j̸=ℓ=1
{gℓ} , gℓ∈ {cℓ, dℓ} .
Then, p|e∈ Pk(e)is uniquely determined by its values in xi := cj+ki(dj− cj)∈ e, i ∈ {0, 1, ..., k}, and has the Lagrangian representation
p|e(x) =
k
∑
i=0
Li(x) p|e(xi) , x∈ e .
(2.4.59) Lemma: A Lagrangian nite element of type [k] is unisolvent. Proof. The proof is an immediate consequence of Lemma 2.4.56.
(2.4.60) Lemma: Let ˆT := [0, 1]d be the quadrilateral reference element and ˆ PTˆ := Qk( ˆT ) , k∈ N , (2.4.61a) ˆ ΣTˆ := {ˆp(ˆx) | ˆx ∈ L[k]( ˆT ) , ˆp∈ ˆQTˆ} . (2.4.61b)
The Lagrangian nite elements of type [k] are a ne equivalent to the refer-ence element ( ˆT , ˆQTˆ, ˆΣTˆ).
Proof. The proof is left as an exercise.
(2.4.62) De nition: A quadrilateral triangulation Th(Ω)is called
geometri-cally conforming, if the intersection of two distinct elements is either empty or consists of a common face or a common edge or a common vertex. (2.4.63) De nition: Let Ω ⊂ Rd be the union of a nite number of
quadri-laterals in Rd and let T
h(Ω) be a quadrilateral triangulation. The nite
element space Vh consisting of quadrilateral nite elements of type [k] is
called a quadrilateral Lagrangian nite element space and will be denoted by S[k](Ω,Th(Ω)), i.e., we have
S[k](Ω,Th(Ω)) := {vh: Ω→ R | vh|T ∈ Qk(T ), T ∈ Th(Ω)} . We further de ne the subspace
S[k],0(Ω,Th(Ω)) := {vh∈ S[k](Ω,Th(Ω))| vh|∂T∩∂Ω = 0, T∈ Th(Ω), T∩∂Ω ̸= ∅} . (2.4.64) Lemma: Let Th(Ω) be a geometrically conforming triangulation of
the computational domain Ω. Then, there holds
S[k](Ω,Th(Ω))⊂ H1(Ω) , (2.4.65a)
S[k],0(Ω,Th(Ω))⊂ H01(Ω) . (2.4.65b)
Proof. The proof is left as an exercise.
(2.4.66) De nition: Let Nh(Ω) ={x ∈ L[k](T )| T ∈ Th(Ω)} , card(Nh(Ω)) =
nh. The function φi∈ S[k](Ω,Th(Ω))as given by (2.4.67) φi(xj) = δij , 1≤ i, j ≤ nh ,
is called the nodal basis function associated with the nodal point xi∈ Nh(Ω).
The set of these functions is a basis of S[k](Ω,Th(Ω)).
The notion 'Lagrangian nite element' stems from the underlying polyno-mial interpolation of Lagrangian type. Another type of polynopolyno-mial interpo-lation is given by Hermite interpointerpo-lation which is based on the interpointerpo-lation of point values and derivatives of a function (cf. Section 2.1.5 in Stoer and Bulirsch (2002)). As far as the construction of Hermitian nite elements and
2.4 Conforming nite element methods 29
associated Hermitian nite element spaces is concerned, we refer to chapter 2.3 in Ciarlet (2002).
The accuracy of nite element approximations uh ∈ Vh of the solution u∈ V of (2.4.1) can be determined by a priori error estimates of the global
discretization error u − uh within the framework of interpolation in Sobolev
spaces.
(2.4.68) De nition: Let Th(Ω)be a simplicial triangulation of Ω ⊂ Rd, and
let (T, PT, ΣT), T ∈ Th(Ω), where PT = Pk(T ) such that dim(PT) = nk and ΣT = {p(xi | xi ∈ Lk(T ), 1 ≤ i ≤ nk} be a simplicial Lagrangian nite element of type (k). Moreover, let φ(i)
T , 1 ≤ i ≤ nk, be the basis functions
associated with the nodal points xi. Then, IT : V ∩ C(T ) → PT as given by
ITv := nk
∑
i=1
v(xi) φ(i)T , v∈ V ∩ C(T )
is called the local interpolation operator. If Vh is the associated simplicial
Lagrangian nite element space, the operator Ih: V ∩ C(Ω) → Vh given by Ihv|T := ITv|T , T ∈ Th(Ω)
is called the global interpolation operator.
Now, if u ∈ V is the solution of (2.4.1) such that u ∈ V ∩ Hk+1(Ω)for
su ciently large k ∈ N such that Hk+1(Ω)⊂ C(Ω), then Ihuis well de ned
and Céa's Lemma (cf. Theorem 2.4.10) implies (2.4.69) ∥u − uh∥1,Ω ≤
C
α ∥u − Ihu∥1,Ω .
In order to derive an upper bound for the interpolation error ∥u−Ihu∥1,Ω, we use that in view of the de nition of Ihand due to PT = Pk(T )⊂ H1(T ), T ∈ Th(Ω),there holds (2.4.70) ∥u − Ihu∥1,Ω = ( ∑ T∈Th(Ω) ∥u − ITu∥21,T )1/2 ,
i.e., we obtain an estimate of the global discretization error via an estimate of the global interpolation error by means of estimates of the local interpolation error ∥u − ITu∥1,T, T ∈ Th(Ω). Due to the a ne equivalence of simplicial Lagrangian nite elements (cf. Lemma (2.4.42)), it su ces to provide such an estimate for the reference element ˆT. Moreover, we need estimates of a ne
transformed functions in Sobolev spaces.
(2.4.71) Lemma: Let ˆT be the simplicial reference element and T = FT( ˆT )
{0, 1}. For ˆv := v ◦ FT there holds ˆv ∈ Hm( ˆT ), and there exists a constant C > 0 only depending on d and m such that
|ˆv|m, ˆT ≤ C ∥BT∥ m|det(BT)|−1/2 |v|m,T , (2.4.72a) |v|m,T ≤ C ∥B−1 T ∥ m |det(BT)|1/2 |ˆv|m, ˆ T . (2.4.72b)
Further, denoting by hT the diameter of T and by ρT the diameter of the
largest ball contained in T , i.e.,
ρT := sup{diam(K) | K is a ball such that K ⊂ T } ,
there holds ∥BT∥ ≤ hT ρTˆ , ∥B−1T ∥ ≤ hTˆ ρT , (2.4.73a) |det(BT)| ≤ |T | | ˆT| . (2.4.73b)
Proof. The proof of (2.4.72a) by taking into account
|Dαv(ˆˆ x)| ≤ ∥Dˆv(ˆx)∥ = sup
∥ξ∥≤1|Dˆv(ˆx)ξ| ,
where Dˆv(ˆx) stands for the total dierential of ˆv, and by means of the chain rule
Dˆv(ˆx)ξ = Dv(x)(Bξ)
as well as the transformation rule for multiple integrals ∫ ˆ T ∥D(v(F ˆx))∥2dˆx = |det(B−1 T )| ∫ T ∥Dv(x)∥2 dx .
The proof of (2.4.72b) can be done analogously. The proofs of (2.4.73a),(2.4.73b)
are left as exercises.
The following Lemma of Bramble-Hilbert is an important tool in the a priori error analysis of nite element methods.
(2.4.74) Theorem: Let Ω ⊂ Rd be a simply connected Lipschitz domain. For k∈ N let ℓ ∈ Hk(Ω)∗ such that Pk(Ω)⊂ Ker(ℓ). Then, there exists a positive
constant C depending on Ω such that for all u ∈ Hk+1(Ω)
(2.4.75) |ℓ(u)| ≤ C ∥ℓ∥Hk(Ω)∗ |u|k+1,Ω .
Proof. For p ∈ Pk(Ω)there holds
