To maintain the best possible degree of convergence, a finite element discretization is proposed and analyzed for both elliptic and parabolic interface problems. The aim of this thesis is to present some results of Galerkin finite element methods for linear elliptic and parabolic interfacial problems.
Problem Description
Parabolic interface problems: We will also consider the following linear parabolic interface problems of the form Model equations of the form involving discontinuous coefficients are sometimes called diffraction problems of the parabolic type.
Notation and Preliminaries
Below we will discuss some preliminary material that will be used frequently in error analysis in the following chapters. A∇w· ∇v+a(x)wv}dx, (1.2.1) satisfies the following boundedness and coercive properties: For φ, ψ ∈ H1(Ω) there exist positive constants C and c, so that.
A Brief Survey on Numerical Methods
They showed that the finite element solution converges to the true solution with optimal speed in L2 and H1 norms in any interior subdomain. Instead of studying the convergence of the interface problem, the authors of [8] studied the finite element approximation to the penalized problem.
Motivation and Objectives
For non-conforming cases, the convergence of finite element solution to the exact solution is still open. In this thesis, we analyzed an unqualified finite element method for both elliptic and time-dependent parabolic interface problems.
Organization of the Thesis
In addition, for the purpose of practical implementation, the effect of numerical quadrature on finite element solution is also analyzed and related optimal error estimates are obtained. Therefore, for the purpose of numerical calculations, we discuss the effect of numerical quadrature on finite element solution and the related optimal order error estimates are also established.
Finite Element Discretization
Based on the fact that the curved triangles closely follow the actual interface Γ, the convergence of the finite element solution is studied by several authors, see [3], [8] and [42]. The finite element approximation is then defined as a function uh ∈Vh such that.
Convergence Results
Note that the assumptions are satisfied since the triangulation Th∗ is constructed such that the interface Γ coincides with the grid lines of Th∗ (cf. Brenner and Scott [12] or Hackbusch [24]). In addition, we must use the following assumptions about Vh, which are now given in quantities.
Effect of Numerical Quadrature
In this chapter we derived some new priority estimates in the modified finite element method for elliptic interface problems in a two-dimensional convex polygonal domain. The aim of this chapter is to study the convergence of fitted finite element solutions to the exact solution for the elliptic interface problem.
Finite Element Discretization and Some Auxiliary Estimates
Since the solutions in question are globally only on H1(Ω), the standard interpolation theory cannot be applied directly. However, taking into account the arguments of [14], it is possible to obtain optimal error bounds for the interpolant Πh (see Remark 2.4 of [14]). Since the interface is of class C2, we can extend the function vi ∈H2(Ωi) to the whole Ω and obtain the function ˜vi ∈H2(Ω) such that ˜vi =vi on Ωi and.
Convergence Analysis
Therefore, the proposed technique will be useful to address the interface issues on non-smooth domain. In this chapter we extend the finite element analysis of elliptic interface problems, discussed in Chapters 2 and 3, to parabolic interface problems. In the theorem below we prove the a priori estimate for the solution u of the interface problem under suitable regularity conditions on f and g.
From the elliptic regularity estimate for the elliptic interface problem (cf. Theorem 1.2.1), it follows. The purpose of the present chapter is to extend the convergence analysis of fitted finite element method for elliptic interface problems to parabolic interface problems. Optimal order error estimates in L2(L2) and L2(H1) norms are shown to hold even if the global regularity of the solution is low.
The Continuous time Galerkin Approximation with Curved Triangles
We now consider the following interface problems: Let wk ∈H01(Ω) be the solution to the interface problem. 4.2.6) Then it immediately follows from the coercivity of operator A that v =w1+w2. Now we are in a position to discuss the continuous-time Galerkin finite element approximation of (4.2.1), which can be formulated as follows: Find uh(t)∈Vh such that. In the last equality we took advantage of the fact that Lhut−uht ∈Vhand has the definition (4.2.12) of the Lh operator.
The Continuous time Galerkin Approximation with Straight Triangles
In this chapter, a discrete-time discontinuous Galerkin (DG) method is used to analyze the fully discrete scheme for parabolic interface problems. Further, the fully discrete solution converges to the exact solution with an optimal speed at the L2(H1) norm if we use straight triangles instead of curved interface triangles. In this method an approximation to the solution is required as a piecewise constant polynomial function in t, which is not necessarily continuous at the nodes of the defining partition.
Furthermore, we have proven the optimal convergence order for the fully discrete solution in the L2(H1) norm when the gridlines approach the interface. Key to the present analysis is the introduction of parabolic dual problems and newly established convergence results for elliptic projection. In section 5.2, some new optimal a priori error estimates are derived for the fully discrete solution when the interfacial triangles are assumed to be curved triangles.
The Discrete time DG Method with Curved Triangles
Now we introduce the interpolant Pk ∈Vhk of u defined by Z. 5.2.4) Now we present the main results of this section in the following theorems. The following stability result of the solution zn of (5.2.5) is very crucial for the convergence analysis. For the H1 norm estimation, as for the L2 norm, we analyze the following discrete auxiliary problem: For 1≤n ≤M, find wn ∈Vhk such that.
The Discrete time DG Method with Straight Triangles
Then the fully discrete finite element approach to the problem (4.3.5) in this case can be stated as follows: Find Un ∈Vhkn, forn M, such that. If we reason as in the derivation of Lemma 5.2.1, we have the stability result of the solution Un satisfying (5.3.2). Now we are in a position to discuss the main result of this section which is stated in the following theorem.
Applying the standard arguments (cf. [14]), we have the following stability result of the solution v satisfying (5.3.5). In this chapter, a finite element discretization independent of the location of the interface is proposed and analyzed for linear elliptic and time-dependent parabolic interface problems. For an inappropriate finite element method, we present error estimates of optimal order in H1 norm and near optimal order in L2 norm for elliptic problems.
Introduction
Furthermore, an attempt has also been made to study the finite element approximation of the following parabolic interface problems. Furthermore, let A(t,·,·) be the bilinear form corresponding to the operator L(t). 6.1.8) The aim of this chapter is to study the convergence of finite element solutions to the exact solutions of elliptic and parabolic interface problems by means of TH-261_BDEKA. Previous works on the inappropriate finite element method for elliptic interface problems can be found in.
Present work not only generalizes the previous works on unfitted finite element method for elliptic interface problems, but also analyzes time-dependent parabolic interface problems for unfitted finite element method. The layout of this chapter is as follows: Section 6.2 is devoted to unfit finite element discretization and states some interpolation and interface approximation properties necessary for the failure analysis. Finally, the time-dependent parabolic interface problems are discussed in Section 6.4 and related error estimates are obtained.
Unfitted Finite Element Discretization
The main tools used in our analysis are Sobolev including inequality, duality arguments, interpolation and interface approximation results. In Section 6.3, the error estimates H1 and near-optimal L2 for elliptic interface problems are obtained. Here, the smooth interface Γ is approximated by the line segment AB, A and B being the intersection points of the interface Γ with the two ends DE and DF in the interface triangles as shown in Figure 6.1.
Let Vh be a family of subspaces of finite elements in H01(Ω) defined on ˜Th, consisting of piecewise linear polynomials that zero on the boundary ∂Ω. According to the proof of Lemma 3.2.1, the following optimal error bounds for the linear interpolant Πh can be obtained. Arguing as in the derivation of Lemma 3.2.2, we obtain the following result, which will be useful for our later analysis.
Convergence Analysis for Elliptic Interface Problems
Let ΓK be the arc common to the interface Γ and the interface triangle K, and ΓKh the constraint of Γh inK ∈ TΓ∗. In view of Lemma 6.2.1, it is sufficient to set bounds on the term Πhu−uh.
Time Dependent Parabolic Interface Problems
The continuous time Galerkin approximation
This subsection is devoted to the continuous-time Galerkin approach to time-dependent parabolic interface problems. We determine error estimates of the optimal order in the L2(H1) norm and the near-optimal order in the L2(L2) norm. The continuous-time Galerkin finite element approximation of (6.1.8) is shown as follows: Find uh : [0, T] → Vh such that uh (0) = Lhu0 and satisfies.
Discrete time Galerkin method
In this chapter, we will present a numerical experiment on one-dimensional test problems to illustrate our theoretical findings. For each example, we calculate the error between the exact solution and the finite element solution in L2 and H1 standards. In this chapter, numerical results for both fitted and unfitted finite element methods are presented.
Tables 7.1 and 7.2 present the convergence of the fitted and unfitted finite element solutions to the exact solutions, respectively.
Example 2
We choose β1 = 12, β2 = 3 and perform a numerical convergence test for the proposed matched and mismatched finite element methods. The finite element discretization in this case is such that the grid lines coincide with the real interface. We used custom finite element discretization to study the convergence of semidiscrete solution to the exact solution.
Further, the error in the norm L2(H1) is shown to be optimal when the finite element discretization is based on right triangulation (cf. Theorem 4.3.1). The standard energy technique is used to derive the optimal error estimate on the norm L2(H1) for the fully discrete case (cf. Theorem 6.4.4). We have applied both adapted and unadapted finite element methods to check the performance of our algorithms.
Extensions and Remarks
In the future, we would like to study the convergence analysis of this problem using finite element methods. Babuˇska, Finite element method for elliptic equations with discontinuous coefficients, Computer Science p. Elliott, A finite element method for solving elliptic equations with Neumann data on a curved boundary using unadjusted meshes, IMA J.
King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Hansbo, An inconvenient finite element method, based on Nitsche's method, for elliptic interface problems, Comput. Raviart, Use of Numerical Integration in Finite Element Methods for Solving Parabolic Equations, Topics in Numerical Analysis, J.J.H.
