Numerical Verification for Elliptic Boundary Value Problem with Nonconforming P_1 Finite Elements

摘要

We propose a numerical method with the nonconforming P_1 FEM to verify the existence of solutions to an elliptic boundary value problem. Formulating the boundary value problem as a fixed-point problem on the sum space of the nonconforming P_1 finite element space with the Sobolev space of 1st order with zero Dirichlet condition, we construct the numerical verification method based on the Schauder fixed-point theorem. We show a constructive inequality for a boundary integral that appears due to the discontinuity of a nonconforming P_1 finite element function. Finally, we present a numerical example to show our proposed method works well.