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SOME INTERFACE PROBLEMS WITH NON-HOMOGENE JUMP CONDITIONS. partially meet the requirements for the degree of. The aim of this thesis is to study higher-order weak Galerkin finite element methods (WG-FEMs) for some interface problems with non-homogeneous jump conditions.

Problem Description

Due to its mathematical complexity and low regularity of solutions, the study of interface problems has remained a major part of mathematical studies until today. The basic effects of an electric field on a biological cell can be described by considering the cell as a conductive body (cytoplasm) surrounded by a dielectric layer (surface membrane) embedded in a more or less conductive medium.

Figure 1.1: Domain Ω and its sub domains Ω 1 , Ω 2 with interface Γ.
Figure 1.1: Domain Ω and its sub domains Ω 1 , Ω 2 with interface Γ.

Preliminaries

Basic Notation

C0m(Ω) is the space of all Cm(Ω) functions with compact support in Ω and C0∞(Ω) is the space of all infinitely differentiable functions with compact support in Ω. In particular, for p = q = 2, the above inequality is known as the Cauchy-Schwarz inequality in Rd.

Background and Motivation

On the other hand, discontinuity in the solution along the interface adds more challenge than one might imagine. A fitted weak Galerkin finite element method is proposed to approximate the voltage of the pulsed electrical model over the physical media involving an electrical interface (surface membrane) and heterogeneous permittivity and a heterogeneous conductivity.

WG Discretization for Interface Problems

Recently, the weak Galerkin finite element method has attracted much attention in the field of numerical partial differential equations. Due to the use of theRTandBDM elements, the weak Galerkin finite element formulation of [143] was limited to the classical finite element separations of triangles (d = 2) and tetrahedra (d = 3).

Figure 1.3: A typical fitted discretization
Figure 1.3: A typical fitted discretization

Proposed Contents of the Thesis

Convergence analysis for the parabolic interface problem through the finite element procedure has been studied by several authors. The objective of this chapter is to propose and analyze the superior weak Galerkin finite element method for parabolic interface problems.

Semidiscrete Approximation

2, we propose the semi-discrete weak Galerkin finite element approximation and derive an important error relation. Let Th be a finite element partition of Ω satisfying the shape regularity assumption as specified in [144]. From Lemma 1.4.1, it is easy to observe that the finite element space Vh0 is a normed linear space with a three-bar norm given by .

As a direct consequence, the following solvability holds for the weak Galerkin finite element scheme (2.2.4). As with the finite element method, we split our error into two components using the intermediate operator. It follows from the definitions of the discrete weak gradient (2.2.3) and the Qh operator and integration by parts that.

Before proceeding further, we present the following results from the previous literature for our subsequent analysis (cf.

Semidiscrete Error Analysis

2.3.8) Further, in view of (2.3.6), this definition can be expressed by saying that Rhv is the weak Galerkin finite element solution of the following elliptic interface problem with exact solution v (cf. Settingwh = (Rhu−Qhu ) )t−(Rhut−Qhut) in the above equation and applying the positivity of the bilinear map a(., .) we obtain. This chapter is devoted to the extension of spatial semidiscrete a priori error analysis to the fully discrete approximation for parabolic interface problem in a convex polygonal domain.

The aim of the present chapter is to extend the convergence analysis of fitted WG-FEMs for elliptic interface problems to parabolic interface problems. In fact, the error analysis in [50] can be extended for the parabolic interface problems to derive optimal error estimation in L2(L2) norm with some more details arguments. The obtained results aim to improve the fully discrete fault analysis of linear parabolic equations on polygonal meshes with Lipschitz interfaces and non-homogeneous jump conditions.

Let Un =Uhn ={U0n, Ubn} ∈Vh0 be the fully discrete approximation of u att=tn which we will define by the following scheme: Given Un−1 in Vh0, we now determine.

Error Analysis for Backward Euler Scheme

The Crank-Nicolson scheme can be defined by the following scheme: If we have Un−1 in Vh0, we now determine that Un∈Vh0 is satisfiable.

Error Analysis for Crank-Nicolson Scheme

The proposed fully discrete finite element scheme can be easily extended to the numerical approximation of the solutions to the following IBVP. Note that for each iteration the spatial mesh size becomes half of the previous mesh size. For each run i, the EOC of a given sequence of L2 norm errors e(i) defined on a sequence of masks of size h(i) by.

The tables also show the convergence behavior of the fully discrete weak Galerkin solutions in the finite time T = 1 with respect to the L2-norm and the discrete H1 norm. It can be seen from these tables that we have achieved the optimal order of convergence in both norms, which confirms the theoretical prediction as proved in the theorems. In both cases, the errors are calculated at time t = 1 and clearly show the second order of convergence in the norm L2 and the first order of convergence in the discrete norm H1.

Weak Galerkin finite element method is proposed for solving wave equation with interface on the weak Galerkin finite element space (Pk(K),Pk−1(∂K),[Pk−1(K)]2).

Table 3.4.1: Numerical results for L 2 -norm error in Example 3.4.1 at final time
Table 3.4.1: Numerical results for L 2 -norm error in Example 3.4.1 at final time

Introduction

The numerical solution of the wave equation is of fundamental importance for the simulation of time-dependent acoustic, electromagnetic or elastic waves. For such wave phenomena, the scalar second-order wave equation often serves as a model problem. For example, an acoustic wave propagating at different speeds in different media is modeled by the second-order wave equation with discontinuous coefficients.

Finite element approximations of the wave equation with interfaces via conformal finite element algorithms placed at the interface are performed. Previous works on FEM for the wave equation with interfaces only considered linear elements and assumed the continuity of the solution along the interfaces. Fully discrete space-time finite element discretizations are based on inverse Euler approximations.

Optimal a priori error estimates for both semi-discrete and fully discrete schemes are proven in L∞(H1) and L∞(L2) standards.

Error Analysis for the Semidiscrete Scheme

Fully Discrete Error Analysis

Let Uhn={U0n, Ubn} ∈Vh0 be the fully discrete approximation of uatt =tn, which we will define using the following scheme: For each n = 1,2,. Time reconstruction U(t) has been introduced to bridge between continuous time error analysis and discrete time error analysis. Recently, the linear immersed finite element method for second-order wave equation in inhomogeneous media has been discussed in [5] with homogeneous jump conditions.

In the previous result, we derived the same estimate of the optimal order of errors for linear elements, assuming utt ∈H1(H3(Ω1)∩H3(Ω2)).

Numerical Results

In our second numerical example, we considered the line interface given by y= 1/2 in the computational domain Ω. Here we observed that the proposed WG-FEM scheme still achieved the optimal rate of convergence in the L2 norm at the final time t = 1 for the WG space of the form (P2(K), P1(∂K),). We also determined some a priori estimates for the semidiscrete solution, which are very crucial for proving the optimal convergence rate of the fully discrete solution.

To begin with, we first recall the problem of the interface of the electrical interface of the form. One of the first finite element methods to address the electrical interface problem was studied by Ammariet al. The correctness of the model interface problem and the correctness of its solutions were established.

A fully discrete finite element scheme based on backward Euler discretization has been proposed for the numerical approximation of the potential distribution.

Table 4.4.1: The history of L 2 error convergence with time step τ = h
Table 4.4.1: The history of L 2 error convergence with time step τ = h

Preliminaries

Semidiscrete Scheme

The well-placedness of the scheme (5.3.1) can be verified from the fact that the finite element space Vh0 is a normed linear space with respect to the triple norm. A simple application of trace inequality (1.4.17) and discrete weak gradient (5.2.3), it is easy to verify that the triple column norm corresponds to follow the discrete H1 norm. Here we have used the fact that ∥Q0u−Qbu∥∂K ≤ ∥Q0u−u∥∂K and standard approximation properties of L2 projection.

To analyze the error, we split our error into two components using the intermediate operator. As a common technique, we try to derive some error equation involving eh, which is crucial for our later analysis. Specifically, we include the jump functions Φ and Ψ in the discrete formulation to avoid the residual in the error equation.

Fully Discrete Method

Now we propose a completely discrete finite element scheme to approach the solution of the interface problem. Let Uh0 = Qhu(0) and Uhn ={U0n, Ubn} ∈Vh0 be the completely discrete approximation of u at t = tn which we will define by means of the following scheme: Given Uh0 ∈Vh0 Uhn ∈Vh0, 1≤n, determine ≤N, satisfactory. The above estimate together with semi-discrete error estimates and (5.3.5) leads to the following fully discrete error estimates.

Numerical Results

In Chapter 2, we presented a priori error estimates for the spatial semi-discrete scheme for the parabolic interface problem. In Chapter 3, we extended the spatially discrete a priori error analysis to a fully discrete approximation for the parabolic interface problem. We obtained the optimal order of convergence in the L2 norm (see Theorem 3.2.2 and Theorem 3.3.2) for both completely discrete schemes.

Finally, we presented some numerical experiments to validate the theoretical estimates for fully discrete method based on Crank-Nicolson schemes. In Chapter 4, we presented a priori error analysis for wave interface problem with non-homogeneous jump conditions along the interface. We derived optimal order of convergence for both semidiscrete scheme and fully discrete scheme in L∞(L2) norm (see Theorem 4.2.1 and Theorem wt51).

We have also consciously the stability of the semidiscrete solution and derive some estimates which are very crucial to prove the optimal convergence rate of the fully discrete solution.

Figure 5.1: Numerical solution (right) and triangulation (left) of Ω for h = 0.167378 with circular interface (Test Example 5.5.1).
Figure 5.1: Numerical solution (right) and triangulation (left) of Ω for h = 0.167378 with circular interface (Test Example 5.5.1).

Extensions and Remarks

  • Numerical results for L 2 -norm error in Example at final time
  • Numerical results for discrete H 1 -norm error in Example 3.4.1 at final time . 38
  • Numerical results for discrete H 1 -norm error in Example 3.4.2 at final
  • The history of L 2 error convergence with time step τ = h
  • The history of L 2 error convergence
  • The history of L 2 error convergence with time step τ = h

Moon, A higher order immersed discontinuous Galerkin finite element method for the acoustic interface problem, Adv. Zhang, On the uniform convergence of the weak Galerkin finite element method for a singly perturbed biharmonic equation, J. Roy, Weak Galerkin finite element methods for parabolic interface problems with inhomogeneous jump conditions, Numer.

Kumar, Weak Galerkin finite element methods combined with the Crank-Nicolson scheme for parabolic interface problems, J. 55], Proposed convergence of the weak Galerkin finite element method for the second-order linear wave equation in heterogeneous media. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopic meshes, Numer.

110], A least-squares-based weak Galerkin finite element method for second-order elliptic equations, SIAM J.

Figure

Figure 1.1: Domain Ω and its sub domains Ω 1 , Ω 2 with interface Γ.
Figure 1.2: A cell comprising of the cell cytoplasm and the cell membrane.
Figure 1.3: A typical fitted discretization
Table 3.4.1: Numerical results for L 2 -norm error in Example 3.4.1 at final time
+7

References

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