A priori error analysis for the finite element approximations to various interface problems arising in biological media
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The main objective of this thesis is to study a priori error analysis of finite element Galerkin methods for some interface problems arising in biological media. Interface problems are often referred to as differential equations with discontinuous coeffcients. The discontinuity of the physical coeffcients is due to the presence of different material properties across the interface. In biological system it is natural to have heterogeneity in the underlying medium as properties of biological media vary between different layers. Due to the presence of discontinuous coeffcients across the interface, interface problems usually lead to non-smooth solutions. Owing toits mathematical complexity and low regularity of its solutions, the study of interface problems has remained a major part of the mathematical study up to the present day. In this thesis we attempt to study the a priori error analysis of some of the interface problems arising in biological media using fitted finite element method. In our first problem, we analyze finite element Galerkin methods applied to pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. Considering the cell to be a conductive body, embedded in a more or less conductive medium, the governing system involves an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. A tted finite element method with straight interface triangles is proposed to approximate the voltage of the pulsed electric model across the physical media. Optimal pointwise-in-time error estimates in L2-norm and H1-norm are shown to hold for semidiscrete scheme even if the regularity of the solution is low on the whole domain. Further, a fully discrete nite element approximation based on Crank-Nicolson scheme is analyzed and related optimal error estimates are derived. Finally, we give numerical examples to verify the theoretical results. We next proceed to the a priori error analysis of non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell-Cattaneo equation on the physical media with discontinuous coe cients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space-time finite element discretizations is based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L1(L2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for non-homogeneous inner structures. Finally, we have extended our analysis to study the dual-phase-lag(DPL) bio heat model in heterogeneous medium. Well-posedness of the model interface problem and a priori estimates of its solutions are established. A new non-standard elliptic type projection operator is introduced to derive optimal convergence result for the semidiscrete solution in L1(L2) norm. The fully discrete space-time finite element discretizations is based on second order in time Newmark scheme. Optimal a priori error estimate for the fully discrete scheme is proved in L1(L2) norm. Finally, numerical results for two dimensional test problems are presented in support of our theoretical findings. Finite element algorithm presented here can contribute to a variety of engineering and medical applications.
Supervisor: Bhupen Deka