A highly accurate discontinuous Galerkin method for solving nonlinear Bratu's problem

Abstract

In this manuscript, we present an innovative approach to address nonlinear Bratu's problem by applying the Discontinuous Galerkin method. Our method accurately computes two-branched numerical solutions of the problem. Depending on the initial guess function, the resulting numerical solution converges either to the upper or to the lower solution of Bratu's problem. We show that our method achieves an optimal convergence rate of order p+1 when p-degree polynomials are used. To confirm the accuracy and efficiency of our proposed method, we present various numerical results. These findings demonstrate the high precision achieved by Discontinuous Galerkin method, enabling the computation of both branches of the solution.