Numerical solution of the Stokes-Brinkman equation by the usage uf the mixed finite element method

Abstract

This thesis works with the Stokes–Brinkman equation, which is the model for simulation of the fluid flow in a fully saturated porous medium. The Stokes–Brinkman equation is a unique equation combining the Darcy and Stokes equations. More precisely, the Darcy equation is used for simulation of fluid flow in porous region and the Stokes equation simulates the fluid flow in void spaces of a porous medium. Both domains (porous and void) have different values of permeability tensor K, which separates one from another in the Stokes–Brinkman equation. In this thesis, the Stokes-Brinkman equation is used for simulation of the fluid flow through various types of porous domains and boundary conditions. Every domain is discretized by mixed finite element methods by means of Q 2 − Q 1 elements, where the stability is verified. The usage of the weak formulation of the Stokes-Brinkman model together with the mixed finite element method leads to the saddle point system. This saddle point system is ill-conditioned by itself, and it further contains the jumps in coef- ficients from K. Such jumps increase ill-conditioning of the solved saddle point system. This thesis suggests to solve the saddle point system by the GMRES method with appropriate types of preconditioning. Most of the presented preconditioners require to solve a system with SPD matrix A from the saddle point system. Generally, this matrix A contains jumps in coefficients and it is hard to find a general preconditioner. Hence some types of SPD approximation of inverse of matrix A are presented and tested. The thesis offers a general comparison of preconditioning techniques with various types of approximation of inverse of matrix A for different examples. Here, also some technique of a posteriori error estimates, which measure the quality of the mesh on various elements, is presented. This technique discovers singularities which can be caused by sudden changes of the fluid flow or by the inner structure of a porous medium.