Numerical verification of solutions for nonlinear elliptic problems using an \(L^\infty\) residual method (Q1378621)

The authors propose an ingenious numerical enclusion method with \(L^\infty\) error bounds for the solutions of the boundary value problem \(-\Delta u=f(u)\) in a convex polygonal subdomain \(\Omega\subset \mathbb{R}^2\) with \(u=0\) on \(\partial\Omega\). The method needs not the assumption of the existence of the solution, but this follows as a result from the estimates. As the main tool, a constructive error estimate for the finite element approximation of the Poisson equation \(-\Delta v=g\) \((g\in L^2(\Omega))\), \(v=0\) on \(\partial \Omega\) is employed. The final results are demonstrated for the equation \(-\Delta u=u^2\).