Jesse Chan

A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms

We introduce a DPG method for convection-dominated diffusion problems. The choice of a test norm is shown to be crucial to achieving robust behavior with respect to the diffusion parameter (Demkowicz and Heuer 2011) [18]. We propose a new inflow boundary condition which regularizes the adjoint problem, allowing the use of a stronger test norm. The robustness of the method is proven, and numerical experiments demonstrate the methods robust behavior.

GPU-accelerated discontinuous Galerkin methods on hybrid meshes

This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/96151

Locally conservative discontinuous Petrov-Galerkin finite elements for fluid problems

We develop a locally conservative formulation of the discontinuous Petrov-Galerkin finite element method (DPG) for convection-diffusion type problems using Lagrange multipliers to exactly enforce conservation over each element. We provide a proof of convergence as well as extensive numerical experiments showing that the method is indeed locally conservative. We also show that standard DPG, while not guaranteed to be conservative, is nearly conservative for many of the benchmarks considered. The new method preserves many of the attractive features of DPG, but turns the normally symmetric positive-definite DPG system into a saddle point problem.

GPU-accelerated Bernstein-Bezier discontinuous Galerkin methods for wave problems

We evaluate the computational performance of the Bernstein-Bezier basis for discontinuous Galerkin (DG) discretizations and show how to exploit properties of derivative and lift operators specific to Bernstein polynomials for an optimal complexity quadrature-free evaluation of the DG formulation. Issues of efficiency and numerical stability are discussed in the context of a model wave propagation problem. We compare the performance of Bernstein-Bezier kernels to both a straightforward and a block-partitioned implementation of nodal DG kernels in a time-explicit GPU-accelerated DG solver. Computational experiments confirm the advantage of Bernstein-Bezier DG kernels over both straightforward and block-partitioned nodal DG kernels at high orders of approximation.

Weight-Adjusted Discontinuous Galerkin Methods: Wave Propagation in Heterogeneous Media

Time-domain discontinuous Galerkin (DG) methods for wave propagation require accounting for the inversion of dense elemental mass matrices, where each mass matrix is computed with respect to a parameter-weighted L2 inner product. In applications where the wavespeed varies spatially at a sub-element scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method which reduces storage costs by replacing the weighted L2 inner product with a weight-adjusted inner product. This equivalent inner product results in an energy stable method, but does not increase storage costs for locally varying weights. A-priori error estimates are derived, and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed.

Weight-Adjusted Discontinuous Galerkin Methods: Curvilinear Meshes

Traditional time-domain discontinuous Galerkin (DG) methods result in large storage costs at high orders of approximation due to the storage of dense elemental matrices. In this work, we propose a weight-adjusted DG (WADG) methods for curvilinear meshes which reduce storage costs while retaining energy stability. A priori error estimates show that high order accuracy is preserved under sufficient conditions on the mesh, which are illustrated through convergence tests with different sequences of meshes. Numerical and computational experiments verify the accuracy and performance of WADG for a model problem on curved domains.

Orthogonal Bases for Vertex-Mapped Pyramids

Discontinuous Galerkin (DG) methods discretized under the method of lines must handle the inverse of a block diagonal mass matrix at each time step. Efficient implementations of the DG method hinge upon inexpensive and low-memory techniques for the inversion of each dense mass matrix block. We propose an efficient time-explicit DG method on meshes of pyramidal elements based on the construction of a semi-nodal high order basis, which is orthogonal for a class of transformations of the reference pyramid, despite the non-affine nature of the mapping. We give numerical results confirming both expected convergence rates and discuss efficiency of DG methods under such a basis.

A Comparison of High Order Interpolation Nodes for the Pyramid

The use of pyramid elements is crucial to the construction of efficient hex-dominant meshes [M. Bergot, G. Cohen, and M. Durufle, J. Sci. Comput., 42 (2010), pp. 345--381]. For conforming nodal finite element methods with mixed element types, it is advantageous for nodal distributions on the faces of the pyramid to match those on the faces and edges of hexahedra and tetrahedra. We adapt existing procedures for constructing optimized tetrahedral nodal sets for high order interpolation to the pyramid with constrained face nodes, including two generalizations of the explicit warp and blend construction of nodes on the tetrahedron [T. Warburton, J. Engrg. Math., 56 (2006), pp. 247--262]. Comparisons between nodal sets show that the lowest Lebesgue constants are given by warp and blend nodes for order

A dual PetrovGalerkin finite element method for the convectiondiffusion equation

N\leq 7

A geometric multigrid preconditioning strategy for DPG system matrices

and Fekete nodes for