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Journal of Computational and Applied Mathematics

journal homepage:www.elsevier.com/locate/cam

Convergence analysis of finite element method for a parabolic obstacle problem

Thirupathi Gudi

^∗

, Papri Majumder

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

a r t i c l e i n f o

Article history:

Received 23 February 2017 Received in revised form 3 July 2018

MSC:

65N30 65N15 Keywords:

Finite element

Parabolic obstacle problem Variational inequalities

a b s t r a c t

A conforming finite element method is proposed and analyzed for numerical approximation of the solution of a parabolic variational inequality of the obstacle problem.

The model problem, which is originally proposed using a general obstacle, is reframed as a model problem with zero obstacle but with non-homogeneous Dirichlet boundary conditions. Subsequently the discrete problem is reframed and the error analysis proving the convergence of the method is performed. The analysis requires a positive preserving interpolation with non-homogeneous Dirichlet boundary condition and a post-processed solution that satisfies the boundary conditions sharply. The results in the article extend the results of (Johnson, SINUM, 1976) for a zero obstacle function to a more general obstacle function.

1. Introduction

The numerical analysis of parabolic variational inequalities has been an active field of research in the past few decades.

The parabolic obstacle problem is one of the well known typical models for studying the parabolic variational inequalities.

The parabolic obstacle problem appears naturally in the American option problem, Stefan problem and in electrochemical machining problem. The numerical analysis of this class of problems has created interest as they naturally provide challenges both in theory and computation. Some studies dealing with error analysis for finite element approximations of parabolic variational inequalities have successfully been done in the literature [1–8]. In [2], the convergence of a truncation method for the numerical solution of the parabolic variational inequality has been obtained with obstacle

ψ ∈

C²(_Ω

¯

_{) and}

ψ ≤

0 on the boundary

∂

Ω, under the assumptionsf

,

^ft

∈

C(

[

0

,

^T

]; ¯

_Ω), wheref is the source term. An L^∞-convergence of an approximation for the parabolic variational inequality with the zero obstacle has been established under the regularity condition u_tt

∈

L²(

[

0

,

^T

];

L²(Ω)) and with a certain assumption on the angle of the triangleT in triangulationT_h in [3]. AnL²-error estimate for a fully-discrete approximation of the solution of parabolic variational inequality with the zero obstacle has been studied in [4]. In that article the fully-discrete scheme is defined by continuous linear finite element for space approximation and general

θ

-scheme (i.e. a general finite difference discretization in time including the backward Euler, forward Euler and Crank–Nicolson scheme) for time discretization. An a posteriori error analysis for parabolic variational inequalities (motivated by American option for baskets) has been done in [5]. In that article the authors gave an upper bound for the error inL²(0

,

^T

;

H¹(Ω)) of fully discrete method (i.e. piecewise linear finite elements in space and the backward Euler method in time). In [6] the authors studied a finite element approximation of a parabolic obstacle problem with nonsmooth initial data for one dimension of spatial variable. An error estimate for

∗ Corresponding author.

E-mail addresses: [email protected](T. Gudi),[email protected](P. Majumder).

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parabolic variational inequalities in the uniform norm has been derived in [7]. In that article, the authors coupled the finite element spatial approximation with a semi-implicit scheme to giveL^∞-asymptotic behavior in the uniform norm using discrete maximum principles. In [8], the authors dealt with the design and analysis of a finite element approximation of the parabolic fractional obstacle problem in bounded domains. This analysis relies on the construction and approximation properties of a positivity preserving interpolant. They discretized the truncation with a backward Euler scheme in time, and use first-degree tensor product finite elements for space. Some error estimations dealing with parabolic variational inequality arising from American option problem, Stefan problem, fluid problem, etc. have been studied in [9–15]. For an example, in [9] the authors established an error estimate for finite element approximations of American option prices under admissible regularity. In [13] the authors derived a priorierror estimate for Stefan problem. For the theory of variational inequalities and their corresponding numerical analysis, we refer to the books [16–23]. Moreover, some related numerical analysis for elliptic, parabolic variational inequalities, PDEs and their solvers may be found in [24–26]. For an example, in [26] the authors derived a posteriorierror estimate for implicit backward Euler approximation of parabolic PDEs in Hilbert space.

In this article, we study the convergence analysis of a finite element method for parabolic obstacle problem with general obstacle, i.e., with obstacle

ψ ∈

H²(Ω^{) and}

ψ |

_∂Ω

≤

0 on boundary

∂

Ω^{, where}Ω

⊂

_R². Here, actually we generalize the analysis in [1]. More explicitly, the results in the article can be described as follows since the generalization of the analysis in [1] to the model problem(2.1)–(2.2)is not straight forward:

•

In [1], the model problem is with zero obstacle function. Even though, the analysis for that case was quite difficult, the generalization to nonzero obstacle requires some more attention. For example, let_K

˜

_hbe the discrete version of K

˜

defined by interpolating the obstacle at the mesh points. Then the interpolation defined in [1] does not preserve K

˜

_h

⊂ ˜

K.

•

Secondly since

∂

^u

/∂

^t is not a continuous function, where u is the exact solution, we cannot use the Lagrange interpolation. Because the error analysis requires the time derivative and interpolation commute. In [1], Johnson used a different approach for defining the interpolation that is valid forH¹(Ω) functions and at the same time it preserves_K

˜

_h

⊂ ˜

K(when

ψ

is identically zero) and further the interpolation ofutherein is an element of_K

˜

_h_when the obstacle is a zero function.

•

In this article, we have non-zero obstacle, therefore the additional difficulties arise due to the nonconforming approximation_K

˜

_h

̸⊂ ˜

K. We rewrite the original problem into a problem with zero obstacle by translation and by introducing nonhomogeneous Dirichlet boundary condition. Then we can use the positivity preserving interpolation (with appropriate modifications at the boundary) in [27,28].

•

In [1], the analysis uses the discrete solution to be inK. But since the nonhomogeneous Dirichlet boundary condition are approximated on the boundary, this cannot hold. We require to define a post-processed solution satisfying the Dirichlet boundary conditions sharply from the discrete solution.

The rest of the article is organized as follows. In Section2, we first introduce some notations and give the continuous setting of the obstacle problem with general obstacle. Then we convert the general obstacle problem into zero obstacle problem with non homogeneous boundary condition by translation. We recall some required known regularity results from [29]. In Section3, we define the discrete problem and define an interpolation operator which satisfies a stability property inL²andH¹. We then derive corresponding interpolation error estimates. Further, we construct a post-processed discrete solution, and derive some error estimate for the discrete solution and post-processed solution. In Section4, we discuss a priori error estimates with these ingredients and derive order of convergenceh

+

(∆^t)³⁴^(log_∆¹_t⁾¹⁴. Finally in Section5, we present numerical experiments to illustrate the theoretical results.

2. Model problem

LetΩ

⊂

_R²be a convex polygonal domain with boundary

∂

Ω. We postpone the 3 dimensional case to future research as there will be some technical difficulties in the analysis, although the ideas could be similar. For a nonnegative integer m

≥

0 and 1

≤

p

≤ ∞

, letW^m^,^p(Ω) be the Sobolev space equipped with the norm

∥ v ∥

_Wm,p(Ω⁾

:=

⎛

⎝

∑

|α|≤m

∫

Ω

|

D^α

v |

^pdx

⎞

⎠

1/p

,

with the standard modification forp

= ∞

. HereD^αdenotes the distribution derivative of order

| α |

. We denote the space byH^m(Ω^{) when}^p

=

2. The spaceH₀¹(Ω) denotes the subspace ofH¹(Ω) with zero trace (vanishing on the boundary in the trace sense).

LetJ

= [

0

,

^T

]

(T

>

0) be the time interval that the problem is defined. IfXis a normed linear space equipped with the norm

∥ · ∥

_X, then define the spacesL^p(J

;

X) for 1

≤

p

≤ ∞

as the set of all functions

w :

J

→

Xsuch that

∥ w ∥

_Lp(J;_X₎

:=

(∫ T 0

∥ w

^(t⁾

∥

^p_Xdt )¹/^p

,

¹

≤

p

< ∞ ,

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and

∥ w ∥

_L^∞_(J;_X₎

:=

sup

t∈J

∥ w

^(t⁾

∥

_X

.

Finally letC(J

;

X) be the space of all continuous functions

w :

J

→

X.

Define a bilinear form abya(

w, v

⁾

=

(

∇ w, ∇ v

). Hereafter, we denote the

[

L²(Ω⁾

]

^d inner product by (

· , ·

), where d

=

1

,

^2.

Let

ψ ∈

W²^,^∞(Ω) be a given function with

ψ |

_∂Ω

≤

0. Define the closed convex set by K

˜ := { v ∈

H₀¹(Ω⁾

: v ≥ ψ

^{a.e. in}Ω

} .

The model problem consists of findingu

:

J

→ ˜

Ksuch that a.e. onJ, (

∂

^u

∂

^t

, v −

u )

+

a(u

, v −

u)

≥

(f

, v −

u)

∀ v ∈ ˜

K

,

^(2.1)

u(x

,

⁰⁾

=

u₀(x) x

∈

_Ω

,

^(2.2)

where

f

∈

C(J

;

L^∞(Ω⁾⁾

, ∂

^f

∂

^t

∈

L²(J

;

L^∞(Ω⁾⁾

,

u₀

∈

W²^,^∞(Ω⁾

∩ ˜

K

.

The model problem(2.1)–(2.2)has a unique solutionuand there holds [29]

u

∈

L^∞(J

;

W²^,^p(Ω⁾⁾

,

¹

≤

p

< ∞ ,

∂

^u

∂

^t

∈

L²(J

;

H₀¹(Ω⁾⁾

∩

L^∞(J

;

L^∞(Ω⁾⁾

,

^(2.3)

(

∂

⁺^u

∂

^t

, v −

u )

+

a(u

, v −

u)

≥

(f

, v −

u)

∀ v ∈ ˜

K

,

^t

∈

J

,

where

∂

⁺^u

/∂

^t denotes the right-hand derivative ofuwith respect tot. Define the set K

:= { v ∈

H¹(Ω⁾

: v ≥

0 a.e. inΩ

, v |

_∂Ω

= − ψ } .

We introduce

w :=

u

− ψ

^{. Then}

w

^satisfies

w :

J

→

Kand (

∂w

∂

^t

, v − w

)

+

a(

w, v − w

⁾

≥

(f

+

_∆

ψ, v − w

⁾

∀ v ∈

K

,

^{a.e. on}^J

,

^(2.4)

w

^(x

,

⁰⁾

=

u₀(x)

− ψ

^(x) ^x

∈

_Ω

.

^(2.5)

It is immediate to see that the model problem(2.4)–(2.5)has a unique solution

w

and there holds

w ∈

L^∞(J

;

W²^,^p(Ω⁾⁾

,

¹

≤

p

< ∞ ,

∂w

∂

^t

∈

L²(J

;

H₀¹(Ω⁾⁾

∩

L^∞(J

;

L^∞(Ω⁾⁾

,

^(2.6)

(

∂

⁺

w

∂

^t

, v − w

)

+

a(

w, v − w

⁾

≥

(f

+

_∆

ψ, v − w

⁾

∀ v ∈

K

,

^t

∈

J

,

where

∂

⁺

w/∂

^tdenotes the right-hand derivative of

w

with respect tot. Since the solutionuof(2.1)–(2.2)satisfies for all t

∈

J (see [29]) that

∂

⁺^u

∂

^t

=

_∆u

+

f a.e. onΩ⁺^(t)

,

∂

⁺^u

∂

^t

=

max

{

f

+

_∆

ψ,

⁰

}

a.e. onΩ⁰^(t)

,

whereΩ⁺^(t)

:= {

x

∈

_Ω

:

u(x

,

^t)

> ψ

^(x)

}

andΩ⁰^(t⁾

:= {

x

∈

_Ω

:

u(x

,

^t⁾

= ψ

^(x)

}

. We conclude that the solution

w

^of (2.4)–(2.5)satisfies

∂

⁺

w

∂

^t

=

_∆

w +

f

+

_∆

ψ

^{a.e. on}Ω⁺^(t⁾

,

^(2.7)

∂

⁺

w

∂

^t

=

max

{

f

+

_∆

ψ,

⁰

}

a.e. onΩ⁰^(t⁾

,

whereΩ⁺^(t⁾

:= {

x

∈

_Ω

: w

^(x

,

^t)

>

⁰

}

andΩ⁰^(t)

:= {

x

∈

_Ω

: w

^(x

,

^t⁾

=

0

}

.

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3. Discrete problem

LetT_hbe a regular simplicial triangulation ofΩ. Denote the set of all vertices ofT_hthat are inΩ^{(resp. on}

∂

Ω^{) by}^V_hⁱ (resp.V_h^b). SetV_h

=

V_hⁱ

∪

V_h^b. Similarly, denote the set of all interior edges ofT_hbyE_hⁱ, the set of all boundary edges byE_h^b, and defineE_h

=

E_hⁱ

∪

E_h^b. Leth_Tbe the diameter ofT, whereTis a triangle ofT_h. Seth

:=

max

{

h_T

:

T

∈

T_h

}

and

|

T

| :=

area ofT. The set of three vertices ofT is denoted byV_T and the length of an edgee

∈

E_his denoted byh_e. Here each triangle T inT_his assumed to be closed. DefineT_hⁱto be the set of all triangles which do not share an edge with boundary

∂

Ω^{. Let} T_h^bdenote the remaining set of triangles, i.e., the set of all trianglesT

∈

T_hwhich have an edgee

∈

E_h^bon its boundary

∂

^T. For simplicity assume that anyT

∈

T_h^bshares at most one edge with boundary

∂

Ω, otherwise the triangleT has no interior node.

Define the discrete space

V_h

:= { v

h

∈

C(_Ω

¯

₎

: v

h

|

_T

∈

_P₁(T)

∀

T

∈

T_h

} ,

wherePr(D) is the set of all polynomials of degree at mostrand defined on an open setD

⊂

_Ω. Define K_h

:= { v

h

∈

V_h

: v

h(z)

≥

0

, ∀

z

∈

V_hⁱ

; v

h(z)

= − ψ

^(z)

, ∀

z

∈

V_h^b

} .

For some positive integerN, let 0

=

t₀

<

^t1

<

^t2

< · · · <

^tN−₁

<

^tN

=

T be a uniform partition ofJ

= [

0

,

^T

]

with

∆^t

=

T

/

^{N, and let}^Jn

=

(t_n−₁

,

^tn

]

. On a discrete set

{ v

⁰

, v

¹

, v

²

, . . . , v

^N

}

ofN

+

1 points, define

∂v

ⁿ

= v

ⁿ

− v

ⁿ⁻¹

∆^t ^forⁿ

=

1

,

²

, . . . ,

^N

.

The fully discrete finite element method for(2.4)–(2.5)is defined as to findW_hⁿ

∈

K_hforn

≥

1 such that (

∂

^W_hⁿ

, v

h

−

W_hⁿ)

+

a(W_hⁿ

, v

h

−

W_hⁿ)

≥

(f(t_n)

+

_∆

ψ, v

h

−

W_hⁿ)

∀ v

h

∈

K_h

,

^(3.1)

W_h⁰

= π

h(u₀

− ψ

⁾

∈

K_h

,

^(3.2)

where

π

his an appropriate interpolation operator.

The following assumption is made as in [1]:

IfD_n

= ∪

_t∈J_n(Ω⁺^(t)

∪

_Ω⁺(t_n))

\

(Ω⁺^(t)

∩

_Ω⁺(t_n)), then there holds for constantC

N

∑

n=₁

m(D_n)

≤

C

,

^(3.3)

wherem(D) is the Lebesgue measure ofD

⊂

_R².

Note: Hereafter, C will denote a positive constant, not necessarily the same at each appearance, which is independent of the parameters∆^t ^and^h.

3.1. Positive preserving interpolation

In this section we modify slightly the positive preserving interpolation operator introduced in [27], to deal with nonhomogeneous Dirichlet boundary conditions. Let

v ∈

H¹(Ω) be such that

v |

_∂Ω

=

g

∈

C(

∂

Ω^{). For any} ^z

∈

V_hⁱ, let

ω

zbe the patch of the vertexzwhich is the union of all the triangles sharingz. Define∆zas the largest disk that can be inscribed in

ω

z with center atz. Then, define

α

z

∈

_Rby

α

z

:=

⎧

⎪⎨

⎪⎩ 1

|

_∆_z

|

∫

∆z

v

^(x)^dx ^if ^z

∈

V_hⁱ

, v

^(z) ^if ^z

∈

V_h^b

,

where

|

_∆_z

|

is the two dimensional Lebesgue measure of the disk∆z. Then define the interpolation

π

h

v ∈

V_hby

π

h

v

^(x)

:=

∑

z∈V_h

α

z

φ

z(x)

,

where

φ

z

∈

V_his the Lagrange basis function associated to the vertexzsatisfying

φ

zi(z_j)

= δ

ij for 1

≤

i

,

^j

≤

dim(V_h)

,

and

δ

ijis the Kronecker delta function. Note that if

v ∈

K, then

π

h

v ∈

K_h. For anyT

∈

T_h, define ST

:= ∪

_z∈V_T

ω

z

.

If all three vertices ofT are insideΩ ^and

v ∈

_P₁(ST), then it is easy to check that

π

h

v ≡ v

^on^T, see [27]. Further if at least one vertex is on

∂

Ω ^and

v ∈

_P₁(ST), we have again

π

h

v ≡ v

^on^T. We prove the stability estimates for

π

h.

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Lemma 3.1. Let

v ∈

H¹(Ω⁾be such that

v |

_∂Ω

=

g

∈

H³^/²(

∂

Ω^{). If T}

∈

T_hwith T

∩ ∂

Ω

= ∅

, then

∥ π

h

v ∥

_L2(T)

≤

C

∥ v ∥

_L2(ST)

,

^(3.4)

∥∇ π

h

v ∥

_L2(T)

≤

C

∥∇ v ∥

_L2(ST)

.

^(3.5)

Further if T

∈

T_hwith T

∩ ∂

Ω

̸= ∅

, then

∥ π

h

v ∥

_L2(T)

≤

C (

∥ v ∥

_L2(ST)

+

h_T max

z∈V_T∩V_h^b

|

g(z)

|

)

,

^(3.6)

∥ π

h

v ∥

_L2(T)

≤

C (

∥ v ∥

_L2(ST)

+

h_T

∥∇ v ∥

_L2(ST)

+

h³_T^/²

∥

g^′

∥

_L2(∂ST∩∂Ω)

)

(3.7)

∥∇ π

h

v ∥

_L2(T)

≤

C (

∥∇ v ∥

_L2(ST)

+

h¹_T^/²

∥

g^′

∥

_L2(∂ST∩∂Ω)

)

.

^(3.8)

Proof. We present the proof in several steps.

Proof of (3.4)and(3.6): Note that ifz

∈

V_hⁱ, then it is easy to show that

| α

z

| ≤

Ch⁻_T¹

∥ v ∥

_L2(S_T)

,

and ifz

∈

V_h^b, then it is clear that

| α

z

| = |

g(z)

|

Since

∥ π

h

v ∥

²

L²(T)

≤

C∑

z∈V_T

| α

z

|

²

∥ φ

z

∥

²

L²(T)

≤

C ∑

z∈V_T

h²_T

| α

z

|

²

,

the proof ofL²-stability in(3.4)and(3.6)follows.

Proof of (3.5): SinceT

∈

T_h withT

∩ ∂

Ω

= ∅

, we have that allz

∈

V_T are inside ofΩ. Then note that for anyc

∈

_R,

π

hc

=

conT and

∥∇ π

h

v ∥

_L2(T)

= ∥∇

(

π

h

v +

c)

∥

_L2(T)

= ∥∇ π

h(

v +

c)

∥

_L2(T)

≤

Ch⁻_T¹

∥ π

h(

v +

c)

∥

_L2(T)

≤

Ch⁻_T¹

∥ v +

c

∥

_L2(ST)

,

where we have used(3.4)in the last step. Now choosecas the integral mean of

v

^on^ST and use scaling argument to derive

∥∇ π

h

v ∥

_L2(T)

≤

C

∥∇ v ∥

_L2(ST)

.

This proves(3.5).

Proof of (3.7): Now suppose that there is somez

∈

V_T such thatz

∈

V_h^b.

Case (i): Let

∂

Ω

∩ ∂

^T be an edge and denote it bye. Sinceg

∈

C(e), there is somez₀

∈

esuch that g(z₀)

=

¹

h_e

∫

e

g(s)ds

.

Note that

|

g(z₀)

| =

h⁻_e¹^/²

∥

g

∥

_L2(e)

=

h⁻_e¹^/²

∥ v ∥

_L2(e)

,

and sinceg

∈

H¹(e), we have thatg is absolutely continuous one. Now for anyz

∈

e,

|

g(z)

| ≤ |

g(z₀)

| +

h¹_e^/²

∥

g^′

∥

_L2(e)

,

whereg^′denotes the tangential derivative ofg one. By the trace inequality

|

g(z)

| ≤

h⁻_e¹^/²

∥ v ∥

_L2(e)

+

h¹_e^/²

∥

g^′

∥

_L2(e)

≤

C(

h⁻_e¹

∥ v ∥

_L2(T)

+ ∥∇ v ∥

_L2(T)

)

+

h¹_e^/²

∥

g^′

∥

_L2(e)

.

^(3.9)

Therefore using this in(3.6), we find

∥ π

h

v ∥

_L2(T)

≤

C (

∥ v ∥

_L2(ST)

+ ∥ v ∥

_L2(T)

+

h_T

∥∇ v ∥

_L2(T)

+

h³_T^/²

∥

g^′

∥

_L2(e)

)

≤

C (

∥ v ∥

_L2(ST)

+

h_T

∥∇ v ∥

_L2(T)

+

h³_T^/²

∥

g^′

∥

_L2(e)

)

.

(6)

Case (ii): If

∂

Ω

∩ ∂

^T is not an edge, then choose an edgee

∈

E_h^bsuch thatz

∈

e. Let_T

˜ ∈

T_hbe a triangle withe

⊂ ∂

_T

˜

_. Repeating the arguments that are used for proving(3.9), we arrive at

|

g(z)

| ≤

C(

h⁻_e¹

∥ v ∥

_L2(_T)˜

+ ∥∇ v ∥

_L2(_T)˜

)

+

h¹_e^/²

∥

g^′

∥

_L2(e)

.

Since_T

˜ ⊂

ST ande

⊂ ∂

^ST

∩ ∂

Ω^{, we have}

|

g(z)

| ≤

C(

h⁻_e¹

∥ v ∥

_L2(ST)

+ ∥∇ v ∥

_L2(ST)

)

+

h¹_e^/²

∥

g^′

∥

_L2(∂ST∩∂Ω)

.

Substituting the estimate forg(z) in(3.6), we obtain(3.7).

Proof of (3.8): For the estimate on

∇ π

h

v

, consider anyc

∈

_Rand

∥∇ π

h

v ∥

_L2(T)

= ∥∇ π

h(

v −

c)

∥

_L2(T)

≤

Ch⁻_T¹

∥ π

h(

v −

c)

∥

_L2(T)

≤

Ch⁻_T¹ (

∥ v −

c

∥

_L2(ST)

+

h_T

∥∇ v ∥

_L2(T)

+

h³_T^/²

∥

g^′

∥

_L2(∂ST∩∂Ω)

)

,

where we have used(3.7)in the last step. Again choosingcas the integral mean of

v

^on^ST and using scaling argument, we find that

∥∇ π

h

v ∥

_L2(T)

≤

C (

∥∇ v ∥

_L2(ST)

+

h¹_T^/²

∥

g^′

∥

_L2(∂^ST∩∂Ω⁾

)

.

Therefore the proof follows. _□

The following approximation properties now follow from the stability and polynomial invariance of

π

h: Theorem 3.2. Let

v ∈

H^m⁺¹(Ω^{) (0}

≤

m

≤

1)be such that

v |

_∂Ω

=

g

∈

H³^/²(

∂

Ω^{). If T}

∈

T_hwith T

∩ ∂

Ω

= ∅

, then

∥ v − π

h

v ∥

_L2(T)

≤

Ch^m_T⁺¹

∥ v ∥

_Hm+1(S_T)

,

^(3.10)

∥∇

(

v − π

h

v

⁾

∥

_L2(T)

≤

Ch^m_T

∥ v ∥

_Hm+1(ST)

.

^(3.11)

Further if T

∈

T_hwith T

∩ ∂

Ω

̸= ∅

, then

∥ v − π

h

v ∥

_L2(T)

≤

C (

h_T

∥ v ∥

_H1(ST)

+

h³_T^/²

∥

g^′

∥

_L2(∂ST∩∂Ω)

)

if

v ∈

H¹(T)

,

^(3.12)

∥ v − π

h

v ∥

_L2(T)

≤

Ch²_T

∥ v ∥

_H2(S_T) if

v ∈

H²(T)

,

^(3.13)

∥∇

(

v − π

h

v

⁾

∥

_L2(T)

≤

C (

∥∇ v ∥

_L2(ST)

+

h¹_T^/²

∥

g^′

∥

_L2(∂ST∩∂Ω)

)

if

v ∈

H¹(T)

,

^(3.14)

∥∇

(

v − π

h

v

⁾

∥

_L2(T)

≤

Ch_T

∥ v ∥

_H2(ST) if

v ∈

H²(T)

.

^(3.15) Proof. In the case ifT

∈

T_hwithT

∩ ∂

Ω

= ∅

, then the estimates are direct consequence of the stability and polynomial invariance of the interpolation operator

π

h. Now letT

∈

T_hwithT

∩ ∂

Ω

̸= ∅

. Then by using(3.7), we find for anyp

∈

_P₁(T) that

∥ v − π

h

v ∥

_L2(T)

= ∥ v +

p

− π

h(

v +

p)

∥

_L2(T)

≤ ∥ v +

p

∥

_L2(T)

+ ∥ π

h(

v +

p)

∥

_L2(T)

≤

C (

∥ v +

p

∥

_L2(ST)

+

h_T

∥∇

(

v +

p)

∥

_L2(ST)

+

h³_T^/²

∥

(g

+

p)^′

∥

_L2(∂ST∩∂Ω)

)

.

If

v ∈

H¹(T), then choosep

∈

_P₀(T) as the integral mean of

v

^on^ST. If

v ∈

H²(T), then choosep

∈

_P₁(T) as the Lagrange interpolation of

v

and complete the proof of theL²-norm estimate. Using similar arguments we can derive the estimates for the derivative. For

∥∇

(

v − π

h

v

⁾

∥

_L2(T)

= ∥∇

(

v +

p

− π

h(

v +

p))

∥

_L2(T)

≤ ∥∇

(

v +

p)

∥

_L2(T)

+ ∥∇ π

h(

v +

p)

∥

_L2(T)

≤

C (

∥∇

(

v +

p)

∥

_L2(ST)

+

h¹_T^/²

∥

(g

+

p)^′

∥

_L2(∂ST∩∂Ω)

)

.

Choosing appropriatepcompletes the proof. □

3.2. Discrete solution with sharp boundary condition

The numerical method in(3.1)–(3.2)can be treated as a nonconforming scheme with respect to the Dirichlet data since the discrete problem uses the approximate Dirichlet data

− ψ

hwhich is not equal to the given data

− ψ

on the boundary

∂

Ω^{, where}

ψ

his the linear Lagrange interpolation of

ψ

. Following the details in [30], we construct an intermediate solution W

˜

_hⁿ(which can be treated as a post-processing ofW_hⁿ) with the help ofW_hⁿ and

ψ

as follows. ForT

∈

T_hⁱ, define_W

˜

n

h to be the same asW_hⁿonT. ForT

∈

T_h^b, first defineW_hⁿ^∗on

∂

^T (the boundary ofT) by

W_hⁿ^∗(x

,

^y)

:=

{

− ψ

^(x

,

^y) ^{if (x}

,

^y)

∈ ∂

Ω

,

W_hⁿ(x

,

^y) ^{if (x}

,

^y)

∈ ∂

^T

\ ∂

Ω

.

^(3.16)

(7)

Fig. 3.1. Heightsh1andh2.

Then define_W

˜

ⁿ

h to be someH¹(T) extension ofW_hⁿ^∗

∈

C(

∂

^T^{) to}^T^{such that}_W

˜

ⁿ

h

|

_∂_T

=

W_hⁿ^∗and_W

˜

ⁿ

h

|

_T

=

W_hⁿ

|

_Tfor allT

∈

T_hⁱ. By the construction_W

˜

n

h

∈

H¹(Ω^{) and}^Whⁿ

− ˜

W_hⁿ

≡

0 on any triangleT

∈

T_hⁱ.

LetT

∈

T_h^bandW_hⁿ^∗

∈

C(

∂

^T) be defined by(3.16). Lete

∈

E_h^bbe such thate

⊂ ∂

^T^{. Let}^z

=

(x

,

y) be an interior point ofT and

ℓ

be the line that is orthogonal toepassing throughz, as it shown inFig. 3.1. Letz₁

=

(x₁

,

^y1) andz₂

=

(x₂

,

^y2) be the points of intersection of

ℓ

^with

∂

^T^{. Let}^h1is the distance betweenzandz₁

;

andh₂is distance betweenzandz₂. Then define_W

˜

ⁿ

h onT by W

˜

_hⁿ(x

,

^y)

:=

^h²^W

n h

∗(x₁

,

^y1)

+

h₁W_hⁿ^∗(x₂

,

^y2)

h₁

+

h₂ for (x

,

^y)

∈

T^◦

,

^(3.17)

whereT^◦is the interior ofT.

LetT

∈

T_h^b. SinceW_hⁿ

|

_T

∈

_P₁(T), we note that W_hⁿ(x

,

^y)

=

^h²^W

n

h(x₁

,

^y1)

+

h₁W_hⁿ(x₂

,

^y2)

h₁

+

h₂ for (x

,

^y)

∈

T^◦

,

and

(_W

˜

ⁿ

h

−

W_hⁿ)(x

,

^y)

=

( h₂

h₁

+

h₂ )

(

ψ

h

− ψ

^)(x1

,

^y1)

.

Since

|

h₂

/

^(h1

+

h₂)

| ≤

1, the following result is immediate, see [30].

Theorem 3.3. Let T

∈

T_h^band define_W

˜

ⁿ

h by(3.17), a linear extension of W_hⁿ^∗ given by(3.16). Let e

∈

E_h^bbe an edge of T and assume

ψ

is such that

ψ |

_e

∈

H¹(e). Then there holds

∥ ˜

W_hⁿ

−

W_hⁿ

∥

_L2(T)

≤

Ch

1

e2

∥ ψ − ψ

h

∥

_L2(e)

,

where C is a positive constant, e

∈

E_h^ban edge of T . Moreover,

∥ ˜

W_hⁿ

−

W_hⁿ

∥

_L2(Ω⁾

≤

C

⎛

⎜

⎝

∑

e∈E_h^b

h_e

∥ ψ − ψ

h

∥

²

L²(e)

⎞

⎟

⎠

1/²

.

FromTheorem 3.3and [30, Theorem 3.2], we have the following corollary. The corollary establishes the difference between the numerical solutionW_hⁿof variational inequality(3.1)–(3.2)and post-processing ofW_hⁿin theL²and theH¹ norms. This corollary plays a crucial rule in the subsequenta priorierror analysis to obtain optimal order of convergence.

Corollary 3.4. For each W_hⁿthere exists_W

˜

ⁿ

h

∈

Ksuch that

∥ ˜

W_hⁿ

−

W_hⁿ

∥

_L2(Ω⁾

≤

Ch²and

∥ ˜

W_hⁿ

−

W_hⁿ

∥

_H1(Ω⁾

≤

Ch

.

4. Error estimate

In this section we estimate the error i.e., the difference between the solution of parabolic variational inequality (2.4)–(2.5)and its numerical approximation as described in Section3. We show that this error estimated in a certain norm, is of orderh

+

(∆^t⁾³⁴(

log_∆¹_t)¹₄ .

(8)

Theorem 4.1(Main Theorem).Suppose

w

is the solution of(2.4)–(2.5)and, f

∈

C(J

;

L^∞(Ω⁾⁾

, ∂

^f

∂

^t

∈

L²(J

;

L^∞(Ω⁾⁾

, w

0

∈

W²^,^∞(Ω⁾

∩

K

,

and(3.3)holds. Let W_hbe the solution of the corresponding discrete problem(3.1). Then, there exists a constant C independent of∆t and h such that

max

n∈{₁,...,N}

∥ w

ⁿ

−

W_hⁿ

∥

_L2(Ω)

+

( _N

∑

n=₁

∥ w

ⁿ

−

W_hⁿ

∥

²

H¹(Ω)∆^t )¹₂

≤

C [

h

+

(∆^t)³⁴ (

log 1

∆^t )¹₄]

,

where

w

ⁿ^{and W}_hⁿis the solution at time t_n.

Proof. Let us define

η

^(t⁾

:= w

^(t)

− π

h

w

^{(t) for}^t

∈

J,eⁿ

:= w

ⁿ

−

W_hⁿ and_e

˜

n

:= w

ⁿ

− ˜

W_hⁿ forn

=

0

, . . . ,

^{N. Then,} eⁿ

− η

ⁿ

= π

h

w

ⁿ

−

W_hⁿ. Now,

(

∂

^eⁿ

,

^eⁿ⁾

+

a(eⁿ

,

^eⁿ⁾

=

(

∂

^eⁿ

, η

ⁿ⁾

+

(

∂

^eⁿ

,

^eⁿ

− η

ⁿ⁾

+

a(eⁿ

, η

ⁿ⁾

+

a(eⁿ

,

^eⁿ

− η

ⁿ⁾

=

(

∂

^eⁿ

, η

ⁿ⁾

+

a(eⁿ

, η

ⁿ⁾

+

(

∂w

ⁿ

, π

h

w

ⁿ

−

W_hⁿ)

−

(

∂

^Whⁿ

, π

h

w

ⁿ

−

W_hⁿ)

+

a(

w

ⁿ

−

W_hⁿ

, π

h

w

ⁿ

−

W_hⁿ)

.

^(4.1) First we put

v = ˜

W_hⁿ

∈

Kandt

=

t_nin the inequality of(2.6)and

v = π

h

w

ⁿ

∈

K_h in(3.1), then by adding these two inequalities, we get

−

(

∂

^Whⁿ

, π

h

w

ⁿ

−

W_hⁿ)

−

a(W_hⁿ

, π

h

w

ⁿ

−

W_hⁿ) (4.2)

≤

(fⁿ

+

_∆

ψ, η

ⁿ⁾

−

(fⁿ

+

_∆

ψ,

_W

˜

ⁿ

h

−

W_hⁿ)

+

(

∂

⁺

w

∂

^t ^(tⁿ⁾

,

_W

˜

ⁿ

h

− w

ⁿ )

+

a(

w

ⁿ

,

_W

˜

ⁿ

h

− w

ⁿ⁾

.

By applying inequality(4.2)in Eq.(4.1)it follows that

(

∂

^eⁿ

,

^eⁿ⁾

+

a(eⁿ

,

^eⁿ⁾

≤

6

∑

j=₁

A^j_n

,

^(4.3)

whereA¹_n

=

(

∂

^eⁿ

, η

ⁿ⁾ A²_n

=

a(eⁿ

, η

ⁿ⁾

A³_n

=

(fⁿ

+

_∆

ψ, η

ⁿ⁾

−

a(

w

ⁿ

, η

ⁿ⁾

+

a(

w

ⁿ

,

_W

˜

ⁿ

h

−

W_hⁿ)

−

(fⁿ

+

_∆

ψ,

_W

˜

ⁿ

h

−

W_hⁿ) A⁴_n

=

(

∂

⁺

w

∂

^t ^(tⁿ⁾

− ∂w

ⁿ

,

^Whⁿ

− w

ⁿ )

A⁵_n

=

(

∂

⁺

w

∂

^t ^(tⁿ⁾

,

_W

˜

ⁿ

h

−

W_hⁿ )

A⁶_n

= −

(

∂w

ⁿ

, η

ⁿ⁾

.

By multiplying(4.3)with∆^tand taking summation on both sides overn

=

1

, . . . ,

^{N, we get}

N

∑

n=₁

(eⁿ

−

eⁿ⁻¹

,

^eⁿ⁾

+

N

∑

n=₁

a(eⁿ

,

^eⁿ⁾∆^t

≤

N

∑

n=₁ 6

∑

j=₁

A^j_n∆^t

=

6

∑

j=₁

E_j

,

^(4.4)

whereE_j

=

N

∑

n=₁

A^j_n∆^t

,

^for^j

=

1

,

²

, . . . ,

⁶

.

Since

2

M

∑

n=₁

(eⁿ

−

eⁿ⁻¹

,

^eⁿ⁾

=

M

∑

n=₁

(eⁿ

−

eⁿ⁻¹

,

^eⁿ

−

eⁿ⁻¹)

+

M

∑

n=₁

(eⁿ

−

eⁿ⁻¹

,

^eⁿ⁻¹⁾

+