Weight-Adjusted Discontinuous Galerkin Methods: Curvilinear Meshes

Abstract

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. arXiv:1608.03836v1 [math.NA] 12 Aug 2016 topologically (vertex, edges, faces, interior); however, the number of nodes exceeds the cardinality of natural approximation spaces on simplices. Additionally, such nodal sets have only been constructed up to degree 4 for tetrahedra.A similar approach is taken for flux reconstruction schemes on simplices, which are closely related to filtered nodal DG methods [23,24] where nodes are taken to be unisolvent quadrature points [25,26]. Unlike nodal sets for mass-lumped simplices, these quadrature points do not contain nodes which lie on the boundary, necessitating an additional interpolation step in the computation of numerical fluxes. However, numerical evidence indicates that co-locating nodes and quadrature points reduces instabilities resulting from the aliasing of spatially varying Jacobians [27], though an analysis of high order convergence and energy stability for curvilinear simplices are open problems.Krivodonova and Berger introduced an inexpensive treatment of curved boundaries for two-dimensional flow problems by modifying the DG formulation on affine triangles [28]. This was extended to wave propagation problems by Zhang in [8], and by Zhang and Tan for elements with non-boundary curved faces in [7]. A theoretical stability and convergence analysis remains to be shown, though numerical results suggest that each of these approaches preserves stability and high order accuracy on curvilinear meshes under the condition that curved triangles are well-approximated by planar triangles. However, sufficiently large differences between curved and planar triangles still result in unstable schemes [8].An alternative treatment addressing increased storage costs of curvilinear DG was addressed by Warburton using the Low-Storage Curvilinear DG (LSC-DG) method [29,30]. Under LSC-DG, the spatial variation of the Jacobian is incorporated into the physical basis functions over each element, resulting in identical mass matrices over each element. Work in [30] also includes a priori estimates for projection errors under the LSC-DG basis, and gives sufficient conditions under which convergence is guaranteed. Furthermore, the DG variational formulation is constructed to be a priori stable for surface quadratures with positive weights, allowing for stable under-integration of high order integrands present for curvilinear elements.In [31], the weight-adjusted DG (WADG)...

A weight-adjusted discontinuous Galerkin method for the poroelastic wave equation: Penalty fluxes and micro-heterogeneities

“…3. An energy stable discontinuous Galerkin formulation for poroelastic wave propgation Energy stable discontinuous Galerkin methods for elastic wave propagation have been constructed based on symmetric formulations of the elastodynamic equations [28], and it is straightforward to extend such discontinuous Galerkin formulations to the symmetric poroelastic system. We assume that the domain Ω is exactly triangulated by a mesh Ω h which consists of elements D k which are images of a reference elementD under the local affine mapping.…”

“…We can now specify a DG formulation for poroelastic wave equation (7). The symmetric hyperbolic system in (7) readily admits a DG formulation based on a penalty flux [32,28]. For the symmetric first order poroelastic wave equation, the DG formulation in strong form is given as…”

Robust approaches to handling complex geometries with Galerkin difference methods

“…To overcome this computational challenge, we propose using the weight-adjusted approach of Chan, Hewett, and Warburton [12]. In this approach the mass matrix is approximated as…”

“…Having to factor the curved mass matrix increases the initialization cost and time-stepping of the method as well as the storage requirements. To address this, we propose using a weight-adjusted mass matrix [12]. In this approach the inverse of the Jacobian weighted mass matrix is approximated in a manner that only requires the inverse of the reference mass matrix and the application of the mass matrix weighted with the reciprocal of the Jacobian determinant.…”

Assessment of Hybrid High-Order methods on curved meshes and comparison with discontinuous Galerkin methods

“…In the former reference, orthonormal bases are obtained by means of a modified Gram-Schmidt procedure to ensure numerical-stability at high-polynomial degrees, while in the latter the same goal is attained by incorporating the spatial variation of the element Jacobian into the physical basis functions. On the other hand, recent works by Chan et al [13,14] rely on reference frame polynomial spaces introducing weight-adjusted L 2 -inner products in order to recover high-order accuracy. HDG has been employed on meshes with curved boundaries, mainly in the context of compressible flow problems [31,28]; eXtended HDG with level-set description of interfaces has been recently investigated by Gurkan et al [27].…”