Sharp \(L^{\infty}\)-error estimates for semilinear elliptic problems with free boundaries (Q1115164)

The author considers the numerical approximation of the semilinear Dirichlet problem \(-\Delta u+\beta(u)=f,\) in \(\Omega u=0\) on \(\partial\Omega\). Here \(\beta\) is a maximal monotone graph in \(R^ 2\) having a singularity at the origin, so that free boundaries for the solution, a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute; thus the discrete maximum principle is avoided. Sharp pointwise error estimates are derived for both the smoothing and the discretization process. An optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global convergence rate. These error estimates for solutions, in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates. Two model examples are discussed: the obstacle problem and a combustion equation.