In this paper we consider higher order shape functions for finite elements on quadrilaterals. Using tensor products of suitable Jacobi polynomials, it can be proved that the corresponding mass and stiffness matrices are sparse with respect to 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. Besides the H1 case also the conformal basis functions for H(Div) and H(Curl) are investigated

Beuchler, Sven

Haubold, Tim

Pillwein, Veronika

English

Institutionelles Repositorium der Leibniz Universität Hannover

Received: 25 June 2021 Accepted: 20 September 2021DOI: 10.1002/pamm.202100200Recursion relations for hp-FEM Element Matrices on quadrilateralsTim Haubold1,∗, Veronika Pillwein2, and Sven Beuchler1,31 Leibniz University Hannover, Institute for Applied Mathematics, Welfengarten 1, D-30167 Hannover2 Johannes Kepler Universität Linz, Research Institute for Symbolic Computation, Altenbergerstraße 69, A-4040 Linz3 DFG Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering – Innovation Across Disciplines),D-30167 HannoverIn this paper we consider higher order shape functions for finite elements on quadrilaterals. Using tensor products of suitableJacobi polynomials, it can be proved that the corresponding mass and stiffness matrices are sparse with respect to polynomialdegree p. Due to the orthogonal relations between Jacobi polynomials the exact values of the entries of mass and stiffnessmatrix can be determined. Using symbolic computation, we can find simple recurrence relations which allow us to computethe remaining nonzero entries in optimal arithmetic complexity. Besides the H1 case also the conformal basis functions forH(Div) and H(Curl) are investigated.© 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.1 Formulation of the problem in H1For a polygonal Lipschitz domain Ω ∈ R2 and given f ∈ L2(Ω), we consider the model problem−div(D gradu)) + κu = f in Ω, u = 0 on ∂Ω, (1)with an elementwise constant diffusion matrix D ∈ R2×2 and coefficient κ. For simple notation we assume a Dirichletboundary condition. Problem (1) is discretized by means of the hp-version of finite elements, see e.g. [1]. Denote by Φ =[ϕ1, . . . , ϕN ] a basis for the finite element space contained in H10 (Ω). Following [2] the shape functions can be separated intothree groups ϕv, ϕe and ϕI using a vertex (v), edge (e), interior (I) ansatz.2 Discretization - Shape functions for H1For the definition of our ansatz functions, Legendre polynomials and their integrals are required. They are defined by theRodriguez formula, see e.g. [3]. More precisely, they and their integrals are given byLn(x) =12nn!dndxn((x2 − 1)n), n ∈ N0, and L̂n(x) =∫ x1Ln−1(t) dt, n ≥ 1,respectively. Let C = (−1, 1)2 be the reference square, see [4] for the triangular case. The H1 interior basis functions on Care given byPn,k(x, y) = L̂n(x)L̂k(y) with their gradients ∇Pn,k(x, y) =(Ln−1L̂k(y)L̂n(x)Lk−1(y).), n, k = 2, . . . , p. (2)In addition the vertex and edge bubbles are defined accordingly, see [1], [5]. By using a Kronecker product formulation, weonly need to evaluate the integrals I(n,m)1 :=∫ 1−1 L̂n(t)L̂m(t) dt, I(n,m)2 :=∫ 1−1 Ln−1(t)Lm−1(t) dt and the mixed integralI(n,m)3 :=∫ 1−1 Ln−1(t)L̂m(t) dt.Corollary 2.1 For the integrals I(n,m)i , i = 1, 2, 3, the following recursion formulas holdI(n+1,m+1)1 =m+ n− 3m+ n+ 3I(n,m)1 , I(n+1,m+1)2 =2m− 12m+ 1I(n,m)2 , and I(n+1,m+1)3 =m+ n− 2m+ n+ 2I(n,m)3 . (3)An extension to the cube is straightforward. Since the H1 basis functions are given byPn,m,k(x, y, z) = P̂(0,0)n (x)P̂(0,0)m (y)P̂(0,0)k (z),we can again factor the integrals and use a Kronecker product formulation. Thus the recursion formulas stay the same.∗ Corresponding author: e-mail haubold@ifam.uni-hannover.deThis is an open access article under the terms of the Creative Commons Attribution License, which permits use,distribution and reproduction in any medium, provided the original work is properly cited.PAMM · Proc. Appl. Math. Mech. 2021;21:1 e202100200. www.gamm-proceedings.com 1 of 2https://doi.org/10.1002/pamm.202100200 © 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.2 of 2 Section 18: Numerical methods of differential equations3 H(Div) and H(Curl) on the squareIn this section, the differential equations −∇∇ · u + κu = f and −∇ × ∇ × u + κu = f for vector valued functions arediscretized. These are H(Div) and H(Curl) model problems, respectively. For ease of notation, only the interior bubblefunctions on the reference element are presented.3.1 H(Div)-conformal functionsFollowing e.g. [6], the basis functions for H(Div) on the reference square are defined asP(I)n,k(x, y) =12(−L̂n(x)Lk−1(y)Ln−1(x)L̂k(y)), P(II)n,k (x, y) =12(L̂n(x)Lk−1(y)Ln−1(x)L̂k(y)), n, k = 2, . . . , p,P(II)1,k (x, y) =(0L̂k(y)), P(II)n,1 (x, y) =(L̂n(x)0), n, k = 2, . . . , p.For the mass matrix one can reuse the recursion formulas (3), but there exists another result.Theorem 3.1 With A,B ∈ {I, II}, let I(A,B)n1,k1,n2,k2 :=∫CP(A)n1,k1(x, y)P(B)n2,k2(x, y) dx dy. For n1, n2, k1, k2 ≥ 3, wehaveI(A,B)n1,k1,n2,k2=n1 + k1 + n2 + k2 + 8n1 + k1 + n2 + k2I(A,B)n1−1,k1−1,n2−1,k2−1 (4)+n1 − k1 + n2 − k2 − 4n1 + k1 + n2 + k2I(A,B)n1−1,k1,n2−1,k2 +−n1 + k1 − n2 + k2 − 4n1 + k1 + n2 + k2I(A,B)n1,k1−1,n2,k2−1.P r o o f. The formula can be found using the Mathematica package Guess [7] and proven using the closed form.For the stiffness matrix one needs to assemble∫CDiv(P(I/II)n1,k1(x, y))Div(P(I/II)n1,k1)(x, y) dx dy. Although the divergenceof the basis functions is rather simple, i.e. DivP (I)n,k(x, y) = 0 and DivP(II)n,k (x, y) = Ln−1(x)Lk−1(y) one can still reuse (3)for a fast assembly.3.2 H(Curl) conformal basis functionsThe two types of basis functions for H(Curl) on the reference square are given byP(I)n,k(x, y) =12(Ln−1(x)L̂k(y)L̂n(x)Lk−1(y)), P(II)n,k (x, y) =12(Ln−1(x)L̂k(y)−L̂n(x)Lk−1(y)), n, k = 2, . . . , p,P(II)1,k (x, y) =(L̂k(y)0), P(II)n,1 (x, y) =(0L̂n(x)), n, k = 2, . . . , p.Again the recursion formulas (3) come in handy. Additionally recursion formula (4) and similar formulas can be used as well.For the stiffness part, we need to solve integrals of the form∫Ccurl(P(I/II)n1,k1(x, y)) curl(P(I/II)n2,k2)(x, y) dx dy. Therefore wefirst compute the curl of the basis functions, i.e. curlP (I)n,k(x, y) = 0 and curlP(II)n,k (x, y) = −Ln−1(x)Lk−1(y). Up to sign,these are the same functions as in the divergence for the H(Div) case. Thus we can compute the H(Curl) in the same way asthe H(Div) case.Acknowledgements The research of the second author was partially funded by the Austrian Science Fund (FWF): W1214-N15, projectDK15. The last author has been supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy withinthe Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453). Open access funding enabled and organized by Projekt DEAL.References[1] C. Schwab, p- and hp-finite element methods, Numerical Mathematics and Scientific Computation (The Clarendon Press, OxfordUniversity Press, New York, 1998), Theory and applications in solid and fluid mechanics.[2] B. Szabó and I. Babuška, Finite element analysis, A Wiley-Interscience Publication (John Wiley & Sons, Inc., New York, 1991).[3] G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71 (CambridgeUniversity Press, Cambridge, 1999).[4] T. Haubold, V. Pillwein, and S. Beuchler, PAMM 19(1), e201900446 (2019).[5] G. E. Karniadakis and S. J. Sherwin, Spectral/hp element methods for CFD, second edition, Numerical Mathematics and ScientificComputation (Oxford University Press, Oxford, 2013).[6] S. Zaglmayr, High Order Finite Elements for Electromagnetic Field Computation, PhD thesis, JKU Linz, 2006.[7] M. Kauers, Guessing Handbook, Tech. Rep. 09-07, Research Institute for Symbolic Computation (RISC).© 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH. www.gamm-proceedings.com 16177061, 2021, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pamm.202100200 by Technische Informationsbibliothek, Wiley Online Library on [17/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

Recursion relations for hp-FEM Element Matrices on quadrilaterals

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Recursion relations for hp-FEM Element Matrices on quadrilaterals

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