Safe Numerical Error Bounds for Solutions of Nonlinear Elliptic Boundary Value Problems
We report on a method for computing enclosures of solutions of second-order nonlinear elliptic boundary
value problems, simultaneously proving the existence of a solution in the enclosing set. The old-fashioned
``monotonicity methods'' are well-suited for this task, but only for a restricted class of problems.
Therefore, we propose a new approach which is based on a suitable fixed-point formulation of the problem
and uses, as an essential ingredient, norm bounds for the inverse of the linearization
of the given problem at some approximate solution $\omega$ which is computed numerically. These
norm bounds are obtained via eigenvalue enclosures.
Primary MSC: 65N15 Error bounds
Secondary MSC: 35J25 Boundary value problems for second-order, elliptic equations, 35J60 Nonlinear PDE of elliptic type
Keywords: computer-assisted proof, elliptic boundary value problem, error bounds