Worst case error bounds for the solution of uncertain Poisson equations with mixed boundary conditions

Given linear elliptic partial differential equations with mixed boundary conditions, with uncertain parameters constrained by inequalities, we show how to use finite element approximations to compute worst case a posteriori error bounds for linear response functionals determined by the solution. All discretization errors are taken into account.

Our bounds are based on the dual weighted residual (DWR) method of Becker and Rannacher (2001), and treat the uncertainties with the optimization approach described in Neumaier (2008).

We implemented the method for Poisson-like equations with an uncertain mass distribution and mixed Dirichlet/Neumann boundary conditions on arbitrary polygonal domains. To get the error bounds, we use a first order formulation whose solution with linear finite elements produces compatible piecewise linear approximations of the solution and its gradient. We need to solve nine related boundary value problems, from which we produce the bounds. No knowledge of domain-dependent a priori constants is necessary.