581 research outputs found
Pre- and postprocessing techniques for determining goodness of computational meshes
Research in error estimation, mesh conditioning, and solution enhancement for finite element, finite difference, and finite volume methods has been incorporated into AUDITOR, a modern, user-friendly code, which operates on 2D and 3D unstructured neutral files to improve the accuracy and reliability of computational results. Residual error estimation capabilities provide local and global estimates of solution error in the energy norm. Higher order results for derived quantities may be extracted from initial solutions. Within the X-MOTIF graphical user interface, extensive visualization capabilities support critical evaluation of results in linear elasticity, steady state heat transfer, and both compressible and incompressible fluid dynamics
Superconvergent interpolatory HDG methods for reaction diffusion equations I: An HDG method
In our earlier work [8], we approximated solutions of a general class of
scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous
Galerkin (Interpolatory HDG) method. This method reduces the computational cost
compared to standard HDG since the HDG matrices are assembled once before the
time integration. Interpolatory HDG also achieves optimal convergence rates;
however, we did not observe superconvergence after an element-by-element
postprocessing. In this work, we revisit the Interpolatory HDG method for
reaction diffusion problems, and use the postprocessed approximate solution to
evaluate the nonlinear term. We prove this simple change restores the
superconvergence and keeps the computational advantages of the Interpolatory
HDG method. We present numerical results to illustrate the convergence theory
and the performance of the method
Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints
We consider time discretization methods for abstract parabolic problems with
inhomogeneous linear constraints. Prototype examples that fit into the general
framework are the heat equation with inhomogeneous (time dependent) Dirichlet
boundary conditions and the time dependent Stokes equation with an
inhomogeneous divergence constraint. Two common ways of treating such linear
constraints, namely explicit or implicit (via Lagrange multipliers) are
studied. These different treatments lead to different variational formulations
of the parabolic problem. For these formulations we introduce a modification of
the standard discontinuous Galerkin (DG) time discretization method in which an
appropriate projection is used in the discretization of the constraint. For
these discretizations (optimal) error bounds, including superconvergence
results, are derived. Discretization error bounds for the Lagrange multiplier
are presented. Results of experiments confirm the theoretically predicted
optimal convergence rates and show that without the modification the (standard)
DG method has sub-optimal convergence behavior.Comment: 35 page
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