AdjointBased, Superconvergent Galerkin Approximations of Linear Functionals
 Citation data:

Journal of Scientific Computing, ISSN: 08857474, Vol: 73, Issue: 23, Page: 644666
 Publication Year:
 2017
 Author(s):
 Publisher(s):
 Tags:
 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 adjointrecovery 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 accuracyenhancing 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 socalled hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for k= 1 , 2 , 3 in onespace dimension and for k= 1 , 2 in twospace 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 pversion of the method, that the rate of exponential convergence of the new approximations is about twice that of J(u).