239 research outputs found
High order discontinuous Galerkin methods on surfaces
We derive and analyze high order discontinuous Galerkin methods for
second-order elliptic problems on implicitely defined surfaces in
. This is done by carefully adapting the unified discontinuous
Galerkin framework of Arnold et al. [2002] on a triangulated surface
approximating the smooth surface. We prove optimal error estimates in both a
(mesh dependent) energy norm and the norm.Comment: 23 pages, 2 figure
Analysis of a mixed discontinuous Galerkin method for the time-harmonic Maxwell equations with minimal smoothness requirements
An error analysis of a mixed discontinuous Galerkin (DG) method with Brezzi
numerical flux for the time-harmonic Maxwell equations with minimal smoothness
requirements is presented. The key difficulty in the error analysis for the DG
method is that the tangential or normal trace of the exact solution is not
well-defined on the mesh faces of the computational mesh. We overcome this
difficulty by two steps. First, we employ a lifting operator to replace the
integrals of the tangential/normal traces on mesh faces by volume integrals.
Second, optimal convergence rates are proven by using smoothed interpolations
that are well-defined for merely integrable functions. As a byproduct of our
analysis, an explicit and easily computable stabilization parameter is given
Residual estimates for post-processors in elliptic problems
In this work we examine a posteriori error control for post-processed
approximations to elliptic boundary value problems. We introduce a class of
post-processing operator that `tweaks' a wide variety of existing
post-processing techniques to enable efficient and reliable a posteriori bounds
to be proven. This ultimately results in optimal error control for all manner
of reconstruction operators, including those that superconverge. We showcase
our results by applying them to two classes of very popular reconstruction
operators, the Smoothness-Increasing Accuracy-Enhancing filter and
Superconvergent Patch Recovery. Extensive numerical tests are conducted that
confirm our analytic findings.Comment: 25 pages, 17 figure
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