A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations

We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second-order coercive linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic 'energy' reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine 'truth-mesh' discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid-flux candidate Lagrange multipliers generated by a K-element coarser 'working-mesh' approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomain-local, symmetric Neumann problems. The technique is illustrated for the convection-diffusion and linear elasticity equations.