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A hierarchical duality approach to bounds for the outputs of partial differential equations. (English) Zbl 0953.76054

Summary: We present a technique for generating lower and upper bounds on outputs which are linear functionals of exact or finite element solutions to symmetric or nonsymmetric coercive linear partial differential equations. The method is based on the construction of an augmented Lagrangian which integrates a quadratic ‘energy’ reformulation of the desired output as the objective to be minimized, with finite-element equilibrium equations and (conforming) ‘hybridized’ intersubdomain continuity conditions. The bounds are then derived by appealing to the associated dual unconstrained max-min problem evaluated for optimally chosen Lagrange multipliers generated by a less expensive approximation, such as a low-dimensional finite-element discretization. As in many a posteriori error estimation techniques, the bound calculation requires only the solution of subdomain-local symmetric (Neumann) problems on the refined ‘true’ mesh. The technique is illustrated in the case of one-dimensional convection-diffusion equation.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76R99 Diffusion and convection
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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