I also take this opportunity to express my gratitude to all the faculty members of the Department of Mathematics, IIT Guwahati for their help on various occasions. In this thesis we attempt to study the a priori error analysis of some of the interface problems that arise in biological media using the fitted finite element method.

## Problem Description

The chapter-by-chapter description of the thesis is presented in the last section of this chapter. The value and the spatial distribution of the transmembrane voltage are of significant interest in the electroporation of the cell membrane.

## Notations and Preliminaries

We first approximate the domain Ω1 with a polyhedral domain Ω1, using a quasi-uniform meshTh1, such that all boundary vertices of Ω1,h lie on the boundary of Ω1. For simplicity of explanation, we write A for A (or Aβ) and Ah for Ah (or Aβh), respectively.

## A Brief Survey on Numerical Methods

To begin, we first present a brief review of the literature on interface-mounted FEMs for the elliptical interface problems. The effect of numerical quadrature on the finite element approximation to the exact solution of the elliptic interface problem has been studied in [35] and derived estimates of the optimal order error in L2 and H1 norms.

## Motivation and Objectives

However, to the best of our knowledge, finite element analysis for the general linear second-order hyperbolic equation with discontinuous coefficients has not been studied earlier. The fully discrete space-time finite element discretizations are based on second order in time Newmark scheme.

## Organization of the thesis

In this chapter, we study a priori error estimates for a spatial semi-discrete scheme for the electrical interface problem. We have 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.

## Introduction

In this chapter, a built-in finite element method for the electrical interface model is proposed and analyzed. A fully discrete finite element approximation based on the Crank-Nicolson scheme is proposed and derives optimal error estimates for the fully discrete solution in both the L∞(H1) and L∞(L2) norms.

## Preliminaries and Auxiliary Results

Regarding the approximation properties of the elliptic projection operator defined in ((2.2.4)(or (2.2.5)), we have the following approximation result: In Section 2.4, we will see that it enables us to set H1( L2) optimal )-norm a priori error estimates and generalize the results of [8].

## Spatially Semidiscrete Error Analysis

FEMs for Electrical Interface Model 28 Now differentiate (2.3.1) with respect to and choose vh =u0h in the resulting equation. For higher order derivatives we differentiate (2.3.1) twice with respect to t and choose vh =u00h in the resulting equation. Now we are in a position to derive the L∞(L2) norm error estimate for the semidiscrete scheme in the following statement.

## Fully Discrete Error Analysis

Due to the low global regularity of the true interface problem solution, it is not straightforward to apply the standard finite element error analysis technique of non-interface problems to interface problems. The present work generalizes the results of [8] under the same regularity assumptions of the true solution. The proposed fully discrete finite element scheme can be easily extended for numerical approximation of solutions with IBVP together with the following jump conditions.

## Numerical Results

In Figure 2.1 we show the exact solution and triangulation of the domain Ω with mesh size h at final time step. In Figure 2.2 we show the exact solution and triangulation of the domain Ω with mesh size h= 0.152069 at final time step. In Figure 2.3 we show the exact solution and triangulation of the domain Ω with mesh size h= 0.125 at final time step.

## Introduction

This chapter is devoted to the a priori error analysis in finite element methods for the hyperbolic heat conduction model problem. In this chapter we propose an appropriate finite element method for the hyperbolic heat equation with discontinuous coefficients. The finite element solution converges for the spatially discrete scheme to the real solution with optimal order in L∞(L2) and L∞(H1) norms.

## Preliminaries

FEMs for Non-Fourier Bio Heat Transfer Model 48 Now we prove the following result which is essential to prove the existence of a strong solution to the interface problem. However, it can be proven that the solution to the interface problem is sufficiently smooth in each individual subdomain Ω1 and Ω2 for smooth data. Since we are not aware of proofs of such local regularities for the interface problem in literature, we will assume additional regularity from you that guarantees the convergence results.

## Auxiliary Projections

The proof of the second inequality is analogous to the above proof, so we omit the details. Let Qh be defined by (3.3.15), then for every v ∈ X˜ there is a positive constant C independent of the mesh parameter h, such that. Let Lh be defined by (3.3.16), then for every v ∈ X˜ there is a positive constant C independent of the mesh size parameter h, such that.

## Spatially Semidiscrete Error Estimate Analysis

FEMs for Non-Fourier Bio Heat Transfer Model 56 This together with H1 stability of L2 projection returns. 3.4.4) In the previous estimate we used the fact that. FEMs for Non-Fourier Bio Heat Transfer Model 58 Following the arguments as in (3.4.4), it is easy to establish this. Then we have the following error equation. 3.4.21) Furthermore, by splitting e(t) into standard ρ and θ arguments, we get.

## Fully Discrete Error Analysis

The following Lemma gives the existence and uniqueness of the fully discrete solution Un of u. To calculate the error between Un and un, it is sufficient to determine the error ξn := unh −Un for 1 ≤ n ≤ N. Once we have estimates for ξn, we can easily obtain the error estimates for en :=Un − un by using the triangle inequality and Theorem 3.4.1 Lemma 3.5.3. The proposed fully discrete finite element scheme can be easily extended for the numerical approximation of the solutions for the following IBVP. together with the following jump conditions [u] = 0, h. b) Crank-Nicolson scheme for parabolic interface problems.

## Numerical Results

For the non-Fourier bio-heat transfer model, we first set the physical coefficients as The second set of physical coefficients corresponding to the classical Pennes bio-heat transfer model is given by. Note that the second set of physical coefficients was chosen to emphasize the fact that our numerical scheme is consistent for the classical Pennes bio-heat transfer model and is clearly depicted in Table 3.2.

## Introduction

This chapter is concerned with a priori error analysis for the spatially semidiscrete scheme for the dual-phase-lag (DPL) bioheat model problem. A new attempt is made to provide both a mathematical and numerical framework for the study of the dual-phase-lag (DPL) problem with the bioheat model. An attempt has been made to adopt a priori error analysis for the general second-order linear hyperbolic interface problem.

## Well-posedness of the Model Interface Problem

It can be seen from Lemma 4.2.1 that the existence of a strong solution depends on the greater regularity of the weak solution, which is the main goal of Theorem 4.2.1. We are well aware of the fact that the speed of convergence of finite element approximations depends on the "smoothness" of the solution. An argument similar to the previous one, and after changing the smoothness condition to u0, v0 ∈ H4(Ω) ∩ H01(Ω) and f ∈ H2(J;H1(Ω)) leads to an improvement in the regularity of the strong solution u.

## Auxiliary Results

In the last inequality, we used the fact that kwkY ≤ Ckv − Qhvk and the stability estimate (4.3.4). 4.3.9) We can now define our new non-standard projection operator of elliptic type, which is crucial for our error analysis. We can follow the proof of Theorem 2.3.1 and Theorem 2.3.2 in Section 2 to derive the following optimal pointwise time error estimates for the newly introduced elliptic type projection operator.

## Spatially Semidiscrete Error Analysis

Semidiscrete Scheme for DPL Bio Heat Model 86 where the positive constant C depends on final timeT. Semidiscrete Scheme for DPL Bio Heat Model 88 Similarly we get 4.4.17) Now we prove the convergence result for the semidiscrete scheme in L∞(L2) norm. Semidiscrete Scheme for DPL Bio Heat Model 90 Then Cauchy-Schwartz inequality, Lemma 3.3.1 and continuity of Bh operator leads to.

## Introduction

This chapter is devoted to the extension of the spatial semi-discrete a priori error analysis to a fully discrete approximation for the problem of a dual phase lag (DPL) biological heat transfer model. Then the fully discrete approximation of the finite element problem is defined as follows: Find Un∈Vh such that. The layout of this chapter is as follows: While Section 5.1 introduces the fully discrete scheme and establishes the existence of its solution, we discuss the convergence behavior of the fully discrete approximation for the interface problem in Section 5.2.

## Fully discrete error analysis

Then, using Cauchy-Schwartz inequality, we obtain coercivity and continuity of the bilinear maps B and A. The proposed fully discrete finite element scheme can be easily extended for the numerical approximation of the solutions for the IBVP associated with the jump conditions. The current analysis allows for the generalization of these works to higher order finite element methods by combining the theory in this work with the analysis in [85].

## Numerical Results

Double phase lag (DPL) bio-heat transfer is characterized by thermal relaxation time τq and phase lag for temperature gradient τT. The second set of physical coefficients corresponding to the thermal wave model of bio-heat transfer is given by. Note that the second set of physical coefficients was chosen to emphasize the fact that our numerical scheme is consistent for the thermal wave model of bio-heat transfer and is clearly depicted in Table 5.3.

## Critical Review of the Results

The optimal order of convergence in both rates L∞(L2) and L∞(H1) is obtained for the semidiscrete solution (see Theorem 3.4.1). The derivation of the associated error for the semidiscrete solution strongly depends on this new non-standard projection operator of elliptic type xu. The fully discrete scheme is the well-known Newmark method for the wave equation when we adopt the particular choice for the parameters in the Newmark scheme (cf.

## Extensions and Remarks

Zhang, Galerkin discontinuous finite element methods for interface problems: a priori and a posteriori error estimates, SIAM J. Sinha, L∞(L2) and L∞(H1) error estimates in the finite element method for linear parabolic interface problems, Numer. Li, Immersed interface finite element methods for elliptic interface problems with inhomogeneous jump conditions, SIAM J.