The discontinuous Galerkin method is a class of finite element methods which use completely discontinuous functions as the numerical solutions and test functions. Stable and convergent discontinuous Galerkin methods have been designed in recent years for various types of partial differential equations, including hyperbolic, parabolic, elliptic equations and equations containing higher order spatial derivatives such as the KdV equations.
The flexibility in the choice of the local spaces and the minimum requirement on the triangulation of the domain make the method especially suitable for adaptivity. The method also has excellent parallel efficiency. Discontinuous Galerkin methods have found rapid applications recently, in such diverse areas as aeroacoustics, electromagnetism, gas dynamics, granular flows, meteorology, oceanography, semiconductor device simulation, turbulent flow, weather forecasting, among many others.
Speakers are named first.
2:15-2:40 Slimane Adjerid, Virginia Tech.
Supeconvergence and error estimation for the discontinuous finite element method
The discontinuous Galerkin method (DGM) is an appealing approach to address
problems having discontinuities, such as those that arise in hyperbolic
conservation laws. The DGM uses a discontinuous finite element basis which
simplifies
2:45-3:10 Shanqin Chen, Weinan E and Chi-Wang Shu, Brown University
The Heterogeneous Multi-Scale Method Based on the Discontinuous
Galerkin Method for Hyperbolic and Parabolic Problems
In this paper we develop a discontinuous Galerkin (DG) method, within the framework of the heterogeneous multi-scale method (HMM), for solving hyperbolic and parabolic multi-scale problems. Hyperbolic scalar equations and systems, and parabolic scalar problems are considered. Error estimates are given for the linear equations and numerical results are provided for the linear and nonlinear problems to demonstrate the capability of the method.
3:15-3:40 Katarina Jegdic,
Department of Mathematics, University of Houston
Convergence of a spacetime discontinuous Galerkin method for Temple class systems
We consider a spacetime discontinuous Galerkin method for one-dimensional systems of hyperbolic conservation laws. We assume that the partitions of the spacetime domain satisfy a causality constraint and that the Galerkin basis consists of piecewise constant functions. In the case of genuinely nonlinear Temple class systems (such as equations of multicomponent chromatography), we show that given a causal spacetime mesh, a unique approximation satisfying local Riemann invariant bounds exist. This enables us to prove convergence of the method to a weak solution.
3:45-4:10 Jennifer Ryan, Computer Science and Mathematics Division, Oak Ridge National Lab and Chi-Wang Shu, Division of Applied Mathematics, Brown University.
Post-processing of the discontinuous Galerkin methods for nonuniform mesh
A post-processing technique based on negative order norm estimates for the discontinuous Galerkin method was previously introduced Cockburn, Luskin, Shu, and Süli. The postprocessor allows improvement in accuracy of the discontinuous Galerkin method for time-dependent linear hyperbolic equations from order k+1 to order 2k+1 for a uniform mesh. This improvement in accuracy was extended to include superconvergence of the derivatives, two space dimensions, multi-domains with different mesh sizes, and variable and discontinuous coefficient hyperbolic equations. The uniform mesh assumption gives the convolution kernel a local form and thus allows for simple implementation using small matrix-vector multiplications. In this talk, we present an extension of this postprocessing technique to include smoothly varying and nonuniform meshes.
8:45- 9:10 Guido Kanschat, Universitat Heidelberg
Multilevel discontinuous Galerkin methods on locally refined meshes
On hierarchical meshes generated by local refinement of a coarse mesh, a multilevel method must be devised carefully in order to maintain optimal complexity. Here, we show that the scheme involving local smoothing is particularly suited for discontinuous Galerkin discretizations of elliptic operators. While the efficiency of the method exhibits independence of the refinement pattern of the grid, the method can still be implemented in a general way. Results will be shown for the interior penalty method as well as for an extension to the LDG method.
9:15-9:40 Ohannes Karakashian, University of Tennessee
A posteriori error estimates and convergent adaptive algorithms for a discontinuous Galerkin method
We present some aposteriori error estimates for an interior penalty discontinuous Galerkin approximation of second order elliptic equations with general boundary conditions. We also present an adaptive refinement/coarsening procedure which is capable of achieving any finite error tolerance in a finite number of steps. While this number depends on the tolerance, we have shown that under certain conditions there is a guaranteed convergence rate.
9:45-10:10 Kening Wang, University of South Carolina
Domain Decomposition Preconditioners for C0 Interior Penalty Methods
We study both two-level additive Schwarz preconditioners and the Bramble-Pasciak-Schatz (BPS) preconditioner that can be used in the iterative solution of the discrete problems resulting from C0 interior penalty methods for fourth order elliptic boundary value problems.
2:00-2:15 Bernardo Cockburn (University of Minnesota), Dominik Schoetzau, (British Columbia) and Guido Kanschat (Heidelberg)
DG methods for the Navier-Stokes equations
We uncover the strage fact that DG methods for the Navier-Stokes equations cannot be locally conservative and energy-stable at the same time. We then propose three new ways of dealing with such a problem. Two of them consist in working with exactly divergence-free velocities: In the first, the exactly divergence-free velocity is obtained by a simple postprocessing; in the second, we work directly with divergence-free velocity spaces but show how to avoid having to actually contruct them. The third approach consists in introducing a new pressure that renders the hyperbolic system associated with the Navier-Stokes equations symmetric.
2:30-2:55 Mark Sussman, Florida State University
A discontinuous spectral element method for the level set equation
Level set methodology is crucially pertinent to tracking moving singular surfaces or thin fronts with steep gradients in the numerical solutions of partial differential equations governing complex flow fields. This methodology must be consistent with the basic solution technique for the partial differential equations. To this end, a discontinuous spectral element approach is developed for level set advection and reinitialization as these methods are becoming increasingly popular for the solution of the fluid dynamic problems. Example computations are provided, which demonstrate the high order accuracy of the method.
3:00-3:25 Ramachandran Nair, NCAR
A Discontinuous Galerkin Atmospheric Shallow Water Model
Correction: wrong abstract posted previously. We consider the development of a shallow water model on the "cubed-sphere" based on discontinuous Galerkin method. The continuous flux-form shallow equations in non-orthogonal curvilinear coordinates are employed. The high-order spatial discretization employs a modal or nodal basis set consisting of Legendre polynomials. Time integration of the model is performed by an explicit Runge-Kutta scheme. The accuracy of the model is evaluated using the standard test problems suggested by Williamson et al. (JCP, 1992). Some of the test results are compared with a spectral element model and other global shallow water models.