A Posteriori Output Bound for Partial Differential Equations Based on Elemental Error Bound Computing

Abstract

An efficient parallel a posteriori output bound procedure for linear functionals of finite element solution of partial differential equations is presented. This procedure is based on independently solving the error bound for finite element solution in local elemental Neumann subproblems. In each subproblem a modified error residual equation which satisfies consistency without needing any complemental conditions is solved for the error bound for the finite element solution. The error bounds for both primal and dual problems are directly used in the output bound which is obtained from optimizing an augmented Lagrangian with a quadratic energy reformulation of the desired output as the objective and finite element equilibrium conditions and interelement continuity requirements as constraints. The algorithm is verified by an example of 2D Poisson problem in the last of the paper.