Smoothing and monotone interations for elliptic differential inclusions

Abstract

Elliptic boundary value problems with discontinuous nonlinearities are embedded into a family of smooth problems. First families of nonlinear regularizations are discussed that guarantee that the solutions of the generated auxiliary problems form uniform lower and upper bounds of solutions of the original discontinuous elliptic boundary value problem. Later these regularizations are approximated by inner iterations using linear boundary value problems. Unlike the regular case here, the linear problems are not collectively bounded. Thus a new stopping rule for the inner iteration is introduced. Finally, the rate of convergence of the outer approximation is estimated and the proposed smoothing technique is combined with some finite element discretization. A numerical example illustrates the proposed monotone smoothing and iteration.