98 research outputs found
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
eXtended hybridizable discontinuous Galerkin for incompressible flow problems with unfitted meshes and interfaces
The eXtended hybridizable discontinuous Galerkin (X-HDG) method is developed for the solution of Stokes problems with void or material interfaces. X-HDG is a novel method that combines the hybridizable discontinuous Galerkin (HDG) method with an eXtended finite element strategy, resulting in a high-order, unfitted, superconvergent method, with an explicit definition of the interface geometry by means of a level-set function. For elements not cut by the interface, the standard HDG formulation is applied, whereas a modified weak form for the local problem is proposed for cut elements. Heaviside enrichment is considered on cut faces and in cut elements in the case of bimaterial problems. Two-dimensional numerical examples demonstrate that the applicability, accuracy, and superconvergence properties of HDG are inherited in X-HDG, with the freedom of computational meshes that do not fit the interfacesPeer ReviewedPostprint (author's final draft
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