This thesis consists of three chapters. In chapter one, we give survey of the numerical methods for solving the diffusion equation subject to the specification of mass, numerical methods combining the numerical solutions of the equivalent integral equations with the backward implicit finite difference method, a numerical schemes for obtaining Pade approximant solutions to the initial boundary value problem for one-dimensional diffusion equation with non-local boundary condition, and efficient parallel technique for one-dimensional time-dependent diffusion equation with two non-local constraints. We include the research background and a brief introduction of diffusion equation. In the section 1.4 of this chapter, here we propose a penalty method for solving the diffusion equation subject to the specification of mass and obtain some error estimates. We also present some numerical results.
In chapter two, we proposed a finite difference method and an iterative method for the L-Shape domain based on the domain decomposition method. We use the coarse mesh on the interface and fine mesh in the two domains. We derive the error estimates and have proved our iterative algorithm has a better convergence rate than the standard method.
In chapter three, we introduced the iterative penalty method for the Stokes equations. We construct the iterative penalty algorithm to allow the use of not very small penalty parameter. We prove the error estimation and presented some numerical results to support our analysis.