Computer assisted proofs of solutions to Nonlinear elliptic partial differential equations

In this article, a numerical method is presented for computer assisted proofs to the existence and uniqueness of solutions to Dirichlet boundary value problems in a certain class of nonlinear elliptic equations. In a weak formulation of the problem, a weak solution is described as a zero point of a certain nonlinear map. Based on Newton-Kantorovich theorem, a numerical existence and local uniqueness of solutions are proved by our proposed method. Some conditions need to be checked numerically. It is shown that all errors of numerical computations such as discretization errors and rounding errors are figured out by numerical computations with result verification. Finally, an illustrative numerical result is presented for showing the usefulness of proposed method.