Adjoint-Based, Superconvergent Galerkin Approximations of Linear Functionals

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Journal of Scientific Computing, ISSN: 0885-7474, Vol: 73, Issue: 2-3, Page: 644-666

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Bernardo Cockburn; Zhu Wang
Springer Nature
Computer Science; Mathematics; Engineering
article description
We propose a new technique for computing highly accurate approximations to linear functionals in terms of Galerkin approximations. We illustrate the technique on a simple model problem, namely, that of the approximation of J(u), where J(·) is a very smooth functional and u is the solution of a Poisson problem; we assume that the solution u and the solution of the adjoint problem are both very smooth. It is known that, if uis the approximation given by the continuous Galerkin method with piecewise polynomials of degree k> 0 , then, as a direct consequence of its property of Galerkin orthogonality, the functional J(u) converges to J(u) with a rate of order h. We show how to define approximations to J(u), with a computational effort about twice of that of computing J(u) , which converge with a rate of order h. The new technique combines the adjoint-recovery method for providing precise approximate functionals by Pierce and Giles (SIAM Rev 42(2):247–264, 2000), which was devised specifically for numerical approximations without a Galerkin orthogonality property, and the accuracy-enhancing convolution technique of Bramble and Schatz (Math Comput 31(137):94–111, 1977), which was devised specifically for numerical methods satisfying a Galerkin orthogonality property, that is, for finite element methods like, for example, continuous Galerkin, mixed, discontinuous Galerkin and the so-called hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for k= 1 , 2 , 3 in one-space dimension and for k= 1 , 2 in two-space dimensions, which show that J(u) converges to J(u) with order hand that the new approximations converges with order h. The numerical experiments also indicate, for the p-version of the method, that the rate of exponential convergence of the new approximations is about twice that of J(u).