Energy Error Estimates and Domain Decomposition Methodology In Finite Element Method

Abstract

Differential equations arise in almost every area of science and engineering for example chemical engineering, aerodynamics, fluid dynamics, material science, astrophysics, economics etc. Day by day since the systems are becoming more and more complicated, the differential equations governing the respective models are also becoming more and more complex and nonlinear in nature. Because of this complexity and nonlinearity, it is very difficult to find the exact solution and we need numerical methods in order to solve the differential equations. In the present work, we will use finite element method to find the approximate solutions of the differential equations. Also, while applying any numerical technique, it becomes very important to estimate the error present in the numerical solution. Therefore, in the present work, due importance has been given to both the a priori and a posteriori error estimates. Since many differential equations arising in the fields like oceanography, elasticity etc. have very large domains and at the same time at many instances, because of the need of immediate solution requirements, it becomes very important to use parallel computing, in order to get an immediate solution. Domain decomposition techniques are highly used for incorporating parallelism. In the present work, a domain decomposition algorithm based on monotone Schwarz iterative process has been discussed in the finite element framework. The whole work has been divided into three chapters.

Chapter 1 introduces the basic concepts of differential equations and the solution methodology. A brief history of finite element methods has been presented and Galerkin finite element method has been explained with the help of examples.

Chapter 2 contains elementary estimates on a priori and a posteriori errors present in the finite element solution in the energy norm. These estimates have been derived for Galerkin finite element discretizations of general linear elliptic operators in the research paper entitled ``A tutorial in elementary finite element error analysis: A systematic presentation of a priori and a posteriori error estimates" by J. Stewart and T.J.R. Hughes. In the present work, these estimates have been derived by taking a particular linear elliptic differential equations.

In Chapter 3, a domain decomposition algorithm based on monotone Schwarz iterative process has been discussed using finite element method for a nonlinear problem. The monotone Schwarz iterative domain decomposition algorithm has been presented in the research paper entitled ``A block monotone domain decomposition algorithm for a semilinear convection-diffusion problem" by Boglaev in the finite difference framework. In the presented work, the method has been discussed in the finite element framework. Bibliography has been presented in the last.