Discontinuous Galerkin Methods for Partial Differential Equations

Day 1: Monday, September 26, 2011 Hybridized DG Method and Mimetic Finite Differences Franco Brezzi IUSS and IMATI-CNR, Pavia Via Ferrata 1, 27100 Pavia [email protected] Abstract: The talk will discuss the relationships between certain variants of Mimetic Finite Differences and the Hybridized version of DG methods for some very simple model problem. The talk will discuss the relationships between certain variants of Mimetic Finite Differences and the Hybridized version of DG methods for some very simple model problem. hp-dG timestepping for time-singular parabolic PDEs and VIs Christoph Schwab ETH Zurich ETH Zentrum, CH 8092 Zurich, Switzerland [email protected] Abstract: hp-dG timestepping for time-singular parabolic PDEs and VIs We extend the hp-dG timestepping from Schoetzau and Schwab (SINUM 2000) to abstract parabolic problems which are singular and degenerate in time. We prove existence and uniqueness of space-time variational solutions of these problems, and establish analytic regularity estimates in countably normed, weighted Sobolev spaces in time. We prove exponential convergence of hp-dG timestepping schemes. Joint work with Oleg Reichmann (ETH SAM). hp-dG timestepping for time-singular parabolic PDEs and VIs We extend the hp-dG timestepping from Schoetzau and Schwab (SINUM 2000) to abstract parabolic problems which are singular and degenerate in time. We prove existence and uniqueness of space-time variational solutions of these problems, and establish analytic regularity estimates in countably normed, weighted Sobolev spaces in time. We prove exponential convergence of hp-dG timestepping schemes. Joint work with Oleg Reichmann (ETH SAM). DG Methods in Coastal Applications Clint Dawson University of Texas at Austin 1 University Station, C0200, Austin, Texas 78712 USA [email protected] Abstract: The talk will focus on the application of DG methods in coastal modeling. We will discuss formulations and numerical results for several problems, including hurricane storm surges, wave modeling, and saltwater intrusion. 1 The talk will focus on the application of DG methods in coastal modeling. We will discuss formulations and numerical results for several problems, including hurricane storm surges, wave modeling, and saltwater intrusion. 1 DG methods for Stokes. A simple preconditioner Donatella Marini University of Pavia Via Ferrata 1, 27100 Pavia [email protected] Abstract: Using DG-discretizations H(div) conforming for the Stokes problem we construct a simple uniform preconditioner for the solution of the final linear system. Using DG-discretizations H(div) conforming for the Stokes problem we construct a simple uniform preconditioner for the solution of the final linear system. Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations Chi-Wang Shu Brown University Division of Applied Mathematics, Brown University, Providence, RI 02912, USA [email protected] http://www.dam.brown.edu/people/shu Abstract: In this work, we introduce a general approach for proving optimal L2 error estimates for the semi-discrete local discontinuous Galerkin (LDG) methods solving linear high order wave equations. The optimal order of error estimates hold not only for the solution itself but also for the auxiliary variables in the LDG method approximating the various order derivatives of the solution. Several examples including the one-dimensional third order wave equation, one-dimensional fifth order wave equation, and multi-dimensional Schrödinger equation are explored to demonstrate this approach. The main idea is to derive energy stability for the various auxiliary variables in the LDG discretization, via using the scheme and its time derivatives with different test functions. Special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy. This is a joint work with Yan Xu. In this work, we introduce a general approach for proving optimal L2 error estimates for the semi-discrete local discontinuous Galerkin (LDG) methods solving linear high order wave equations. The optimal order of error estimates hold not only for the solution itself but also for the auxiliary variables in the LDG method approximating the various order derivatives of the solution. Several examples including the one-dimensional third order wave equation, one-dimensional fifth order wave equation, and multi-dimensional Schrödinger equation are explored to demonstrate this approach. The main idea is to derive energy stability for the various auxiliary variables in the LDG discretization, via using the scheme and its time derivatives with different test functions. Special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy. This is a joint work with Yan Xu. Trefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations Ilaria Perugia University of Pavia Department of Mathematics, Via Ferrata 1, 27100 Pavia, Italy [email protected] http://www-dimat.unipv.it/perugia/ Abstract: Several finite element methods used in the numerical discretization of wave problems in frequency domain are based on incorporating a priori knowledge about the differential equation into the local approximation spaces by using Trefftz-type basis functions, namely functions which belong to the kernel of the considered differential operator. These methods differ form one another not only for the type of Trefftz basis functions used in the approximating spaces, but also for the way of imposing continuity at the interelement boundaries: partition of unit, least squares, Lagrange multipliers or discontinuous 2 Several finite element methods used in the numerical discretization of wave problems in frequency domain are based on incorporating a priori knowledge about the differential equation into the local approximation spaces by using Trefftz-type basis functions, namely functions which belong to the kernel of the considered differential operator. These methods differ form one another not only for the type of Trefftz basis functions used in the approximating spaces, but also for the way of imposing continuity at the interelement boundaries: partition of unit, least squares, Lagrange multipliers or discontinuous 2 Galerkin techniques. In this talk, the construction of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations, together with their abstract error analysis, will be presented. This analysis requires new stability estimates and regularity results for the continuous problem which can be of interest on their own. The particular case where the approximating Trefftz spaces are made of plane waves will be considered, and explicit error estimates will be given. These results have been obtained in collaboration with Ralf Hiptmair and Andrea Moiola form ETH Zürich. hp DGFEM in polyhedra II. Exponential convergence Thomas Wihler University of Bern Sidlerstrasse 5, CH-3012 Bern [email protected] http://www.math.unibe.ch/content/personal/e7646/e7813/index ger.html/ Abstract: In this talk we will present and analyze hp-type interior penalty discontinuous Galerkin discretizations for the numerical approximation of linear second-order elliptic PDE in polyhedral domains in 3-d. Particularly, the well-posedness of hp-DGFEM and the exponential convergence in terms of the number N of degrees of freedom of hp-DGFEM shall be established. Here, due to the presence of edge singularities in the exact solution, anisotropic geometric mesh refinements and anisotropic polynomial degree distributions are necessary to achieve exponential convergence rates for hp-DGFEM. To account for the anisotropic regularity of solutions in different (local) coordinate directions, our analysis will be based upon possibly irregular hexahedral meshes. For this type of mesh, hp-DG methods constitute a particularly practical choice. Joint work with D. Schötzau (UBC) and Ch. S̃chwab (ETH Zürich). In this talk we will present and analyze hp-type interior penalty discontinuous Galerkin discretizations for the numerical approximation of linear second-order elliptic PDE in polyhedral domains in 3-d. Particularly, the well-posedness of hp-DGFEM and the exponential convergence in terms of the number N of degrees of freedom of hp-DGFEM shall be established. Here, due to the presence of edge singularities in the exact solution, anisotropic geometric mesh refinements and anisotropic polynomial degree distributions are necessary to achieve exponential convergence rates for hp-DGFEM. To account for the anisotropic regularity of solutions in different (local) coordinate directions, our analysis will be based upon possibly irregular hexahedral meshes. For this type of mesh, hp-DG methods constitute a particularly practical choice. Joint work with D. Schötzau (UBC) and Ch. S̃chwab (ETH Zürich).