The thesis is concerned with superconvergent approximations of linear functionals.
We extend the adjoint error correction technique of Pierce and Giles [SIAM Review,
42 (2000), pp. 247-264] for obtaining superconvergent approximations of functionals
to Galerkin methods. We illustrate the technique in the framework of discontinuous
Galerkin methods for problems in one dimension and two dimensions. In one dimension
our focus is on ordinary differential and convection-diffusion equations. It is well
known that approximations to linear functionals obtained by discontinuous Galerkin
methods with polynomials of degree k can be proven to converge with order 2 k + 1
and 2 k for ordinary differential and convection-diffusion equations, respectively. In
contrast, the order of convergence can be proven to be 4 k + 1 and 4 k, using our
technique. Since both approaches have a computational complexity of the same order,
the adjoint error correction method is clearly a competitive alternative. In two
dimensions we deal with a simple second-order elliptic model problem. We show that
approximate functionals converge with order 4 k with our method. Numerical results
which confirm the theoretical predictions are presented.
University of Minnesota Ph.D. dissertation. February 2010. Major: Mathematics. Advisor: Prof. Bernardo Cockburn. 1 computer file (PDF); v, 63 pages.
Adjoint recovery of superconvergent linear functionals from Galerkin approximations..
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