Efficient Numerical Treatment of High-contrast Diffusion Problems

This dissertation concerns efficient numerical treatment of the elliptic partial differential equations with high-contrast coefficients. High-contrast means that the ratio between highest and lowest values of the coefficients is very high, or even infinite. A finite-element discretization of such equations yields a linear system with an ill-conditioned matrix which leads to significant issues in numerical methods. The research in second chapter introduces a procedure by which the discrete system obtained from a linear finite-element discretization of the given continuum problem is converted into an equivalent linear system of a saddle point type. Then a robust preconditioner for the Lancsoz method of minimized iterations for solving the derived saddle point problem is proposed. Numerical experiments demonstrate effectiveness and robustness of the proposed preconditioner and show that the number of iterations is independent of the contrast and the discretization size. The research in third chapter concerns the case of infinite-contrast problems with almost touching injections. The Dirichlet-Neumann domain decomposition algorithm yields a Schur complement linear system. The issue is that the block corresponding to the highly-dense part of the domain is impossible to obtain in practice. An approximation of this block is proposed by using a discrete Dirichlet-to-Neumann map. The process of construction of a discrete map together with all its properties is described and numerical illustrations with comparison to the solution obtained by the direct method are provided.