Sparsity Optimized High Order Finite Element Functions on Simplices

Beuchler, Sven; Pillwein, Veronika; Schöberl, Joachim; Zaglmayr, Sabine

doi:10.1007/978-3-7091-0794-2_2

Cited by 17 publications

(12 citation statements)

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“…For the triangle and tetrahedron, our construction is based on the concept of scaled polynomials as described by Karniadakis and Sherwin (1999), Schöberl and Zaglmayr (2005), Zaglmayr (2006) and the subsequent work of Beuchler et al (2012a). See also Beuchler and Schöberl (2006); Beuchler and Pillwein (2007); Beuchler et al (2012b) and Beuchler et al (2013) for more details on obtaining good sparsity properties via appropriate selection of Jacobi polynomials.…”

Section: Previous Workmentioning

confidence: 99%

Orientation embedded high order shape functions for the exact sequence elements of all shapes

Fuentes

Keith

Demkowicz

et al. 2015

Computers & Mathematics with Applications

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A unified construction of high order shape functions is given for all four classical energy spaces (H 1 , H(curl), H(div) and L 2 ) and for elements of "all" shapes (segment, quadrilateral, triangle, hexahedron, tetrahedron, triangular prism and pyramid). The discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element. The shape functions are conforming, hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes. Expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape. The polynomial order is allowed to differ for each separate topological entity (vertex, edge, face or interior) in the mesh, so the shape functions can be used to implement local p adaptive finite element methods. Each topological entity may have its own orientation, and the shape functions can have that orientation embedded by a simple permutation of arguments.i

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Section: Previous Workmentioning

confidence: 99%

Orientation embedded high order shape functions for the exact sequence elements of all shapes

Fuentes

Keith

Demkowicz

et al. 2015

Computers & Mathematics with Applications

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“…Warburton et al [12] observed that the stiffness matrix for C 0 basis functions defined using Jacobi polynomials with a specific choice of parameters yielded very sparse and banded blocks in the discretization of the Laplacian. Similarly, Beuchler et al [13,14] proposed using integrated Jacobi polynomials to construct topological basis functions that yield sparse and nearly diagonal stiffness matrices on simplicial meshes. These bases rely on orthogonality properties to sparsify global discretization matrices; in contrast, we leverage properties of Bernstein-Bezier basis functions to simultaneously sparsify elemental derivative operators and factorize elemental lift operators to enable GPU friendly implementations.The paper is structured as follows -we review the formulation of DG methods for the first order wave equation in Section 2.…”

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confidence: 99%

GPU-Accelerated Bernstein--Bézier Discontinuous Galerkin Methods for Wave Problems

Chan¹,

Warburton²

2017

SIAM J. Sci. Comput.

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Abstract. 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.1. Introduction. In early work, Klöckner et al.[1] described how to implement high-order Lagrange discontinuous Galerkin (DG) finite element discretizations [2] of Maxwell's equations on graphics processing units (GPUs). In this paper, we examine how the choice of high-order Lagrange elements necessitates operations with block dense matrices that create computational bottlenecks, limiting computational throughput on GPUs. We address this issue by investigating on the one hand block-partitioned implementations of kernels involving dense matrix multiplication and on the other hand using a Bernstein-Bezier basis, that induces blocked discretization matrices with sparse non-zero blocks. The former is motivated by GPU memory latency [3,4] and strategies adopted by optimized linear algebra libraries [5], while the latter is motivated by recent work on efficient algorithms for high order finite element discretizations.Bernstein polynomials have been known for over a century (see [6] for a survey and retrospective), and are widely used in graphics rendering and computer aided design. However, the study of Bernstein-Bezier polynomials has only recently become an active area of research for high-order finite element discretizations. Both Ainsworth et al. [7,8] and Kirby and Thinh [9, 10] utilized properties of Bernstein-Bezier polynomials to reduce the operational complexity of stiffness matrix assembly and evaluation of stiffness matrix-vector products for high-order continuous finite element discretizations. Kirby later introduced efficient algorithms for adopting Bernstein-Bezier polynomials in discontinuous Galerkin time-domain discretizations of first order symmetric hyperbolic PDEs [11]. These algorithms have complexity that is asymptotically optimal in the number of degrees of freedom. We extend that work by exposing additional intrinsic structure of Bernstein-Bezier derivative and lift operators to enable an asymptotically optimal implementation of the former and a reduced complexity implementation for the latter.A number of bases that yield block sparse finite element matrices have been proposed in the literature. Warburton et al. [12] observed that the stiffness matrix for C 0 basis functions defined using Jacobi polynomials with a specific choic...

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“…We refer to [6] for generalizations to the three dimensional case and to [9] for other PDE's like the Maxwell equations. Similar recursion formulas to the shown formula can be found for the interior block of the mass matrix as well.…”

Section: Discussionmentioning

confidence: 99%

Symbolic Evaluation of hp‐FEM Element Matrices

Haubold

Pillwein

Beuchler

2019

Proc Appl Math and Mech

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In this paper we consider higher order shape functions for finite elements on a triangle. On the reference element Dubiner-like ansatz functions based on suitable integrated Jacobi polynomials are chosen. It can be proved that the corresponding mass and stiffness matrices are sparse for all polynomial degree p. Due to the orthogonal relations between Jacobi polynomials the exact values of the entries of mass and stiffness matrix can be determined. Using symbolic computation, we can find simple recurrence relations which allow us to compute the remaining nonzero entries in optimal arithmetic complexity.

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Sparsity Optimized High Order Finite Element Functions on Simplices

Cited by 17 publications

References 50 publications

Orientation embedded high order shape functions for the exact sequence elements of all shapes

Orientation embedded high order shape functions for the exact sequence elements of all shapes

GPU-Accelerated Bernstein--Bézier Discontinuous Galerkin Methods for Wave Problems

Symbolic Evaluation of hp‐FEM Element Matrices

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