Adjoint-Based, Superconvergent Galerkin Approximations of Linear Functionals

作者：Cockburn Bernardo^*; Wang Zhu

来源：Journal of Scientific Computing, 2017, 73(2-3): 644-666.

DOI：10.1007/s10915-017-0507-7

• Approximation of linear functionals
• Adjoint-based error correction
• Galerkin methods
• Filtering
• Convolution

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 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 is the approximation given by the continuous Galerkin method with piecewise polynomials of degree , then, as a direct consequence of its property of Galerkin orthogonality, the functional converges to J(u) with a rate of order . We show how to define approximations to J(u), with a computational effort about twice of that of computing , which converge with a rate of order . 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 in one-space dimension and for in two-space dimensions, which show that converges to J(u) with order and that the new approximations converges with order . 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 .

• 出版日期2017-12