Verified Solutions of Large Sparse Linear Systems Arising from 3D Poisson Equation

We are concerned with verified numerical solutions of linear systems arising from 3D Poisson equation. By discretizing the equation by the finite difference method or the finite element method, we obtain a sparse linear system with a coefficient matrix being expected to be an M-matrix. To solve such a linear system, iterative solution methods such as the conjugate gradient (CG) method are frequently used. We usually measure a residual norm for checking the convergence. However, we do not know the accuracy of numerical solutions. In this talk, we propose a verification method calculating error bounds of numerical solutions. We discuss both the quality of the verified error bounds and the speed of the verification process. Numerical results will be presented. This is a joint work with Prof. Kengo Nakajima.