Multiscale Partition of Unity Patrick

Abstract. We introduce a new Partition of Unity Method for the numerical homog-enization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved in-dependent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the par-tition of unity property. This precomputation involves the solution of independent prob-lems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for ‘cheap ’ parameter choices