145 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
Analysis of a space--time hybridizable discontinuous Galerkin method for the advection--diffusion problem on time-dependent domains
This paper presents the first analysis of a space--time hybridizable
discontinuous Galerkin method for the advection--diffusion problem on
time-dependent domains. The analysis is based on non-standard local trace and
inverse inequalities that are anisotropic in the spatial and time steps. We
prove well-posedness of the discrete problem and provide a priori error
estimates in a mesh-dependent norm. Convergence theory is validated by a
numerical example solving the advection--diffusion problem on a time-dependent
domain for approximations of various polynomial degree
Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations
We present optimal preconditioners for a recently introduced hybridized
discontinuous Galerkin finite element discretization of the Stokes equations.
Typical of hybridized discontinuous Galerkin methods, the method has
degrees-of-freedom that can be eliminated locally (cell-wise), thereby
significantly reducing the size of the global problem. Although the linear
system becomes more complex to analyze after static condensation of these
element degrees-of-freedom, the pressure Schur complement of the original and
reduced problem are the same. Using this fact, we prove spectral equivalence of
this Schur complement to two simple matrices, which is then used to formulate
optimal preconditioners for the statically condensed problem. Numerical
simulations in two and three spatial dimensions demonstrate the good
performance of the proposed preconditioners
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