arXiv subjects

John Singler

Publications and source records attributed to John Singler.

3 recordsLinked to original sources

arXiv2019

New proper orthogonal decomposition approximation theory for PDE solution data

In our previous work [Singler, SIAM J. Numer. Anal. 52 (2014), no. 2, 852-876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection errors and error bounds for POD reduced order models of PDEs. In this work, we improve on our earlier results concerning POD projections by extending to a more general framework that allows for non-orthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD data approximation errors, and also prove new pointwise convergence results and error bounds for POD projections. We consider both the discrete and continuous cases of POD. We also apply our results to several example problems, and show how the new results improve on previous work.

math.NA↗

arXiv2019

Superconvergent interpolatory HDG methods for reaction diffusion equations I: An HDG$_{k}$ 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.

math.NA↗

arXiv2019

On the superconvergence of a hydridizable discontinuous Galerkin method for the Cahn-Hilliard equation

We propose a hydridizable discontinuous Galerkin (HDG) method for solving the Cahn-Hilliard equation. The temporal discretization can be based on either the backward Euler method or the convex-splitting method. We show that the fully discrete scheme admits a unique solution, and we establish optimal convergence rates for all variables in the $L^2$ norm for arbitrary polynomial orders. In terms of the globally coupled degrees of freedom, the scalar variables are superconvergent. Another theoretical contribution of this work is a novel HDG Sobolev inequality that is useful for HDG error analysis of nonlinear problems. Numerical results are reported to confirm the theoretical convergence rates.

math.NA↗