Modular hp-FEM system HERMES and its application to Maxwell’s equations. (English) Zbl 1157.78356

Vejchodský, Tomáš; Šolín, Pavel; Zítka, Martin

Math. Comput. Simul. 76, No. 1-3, 223-228 (2007).

Summary: We introduce a multi-physics modular \(hp\)-FEM system HERMES. The code is based on a novel approach where the finite element technology (mesh processing and adaptation, numerical quadrature, assembling and solution of the discrete problems, a-posteriori error estimation, etc.) is fully separated from the physics of the solved problems. The physics is represented via simple modules containing PDE-dependent parameters as well as hierarchic higher-order finite elements satisfying the conformity requirements imposed by the PDE. After describing briefly the modular structure of HERMES and some of its functionality, we focus on its application to the time-harmonic Maxwell’s equations. We present numerical results which illustrate the capability of the \(hp\)-FEM to reduce both the number of degrees of freedom and the CPU time dramatically compared to standard lowest-order FEM.

MSC:

78M10	Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

\(hp\)-FEM; time-harmonic Maxwell’s equations; hierarchic higher-order edge elements

Software:

Triangle; UMFPACK; par2Dhp; PETSc; Trilinos

References:

[1]	M. Ainsworth, B. Senior, Aspects of an hp-adaptive finite element method: adaptive strategy, conforming approximation and efficient solvers, Technical Report 1997/2, Department of Mathematics and Computer Science, University of Leicester, England, 1997. · Zbl 0906.73057
[2]	Babuška, I.; Gui, W., The h, p and hp-versions of the finite element method in 1 dimension—part I. the error analysis of the p-version, Numer. math., 49, 577-612, (1986) · Zbl 0614.65088
[3]	Babuška, I.; Gui, W., The h, p and hp-versions of the finite element method in 1 dimension—part II. the error analysis of the h and hp-versions, Numer. math., 49, 613-657, (1986) · Zbl 0614.65089
[4]	Babuška, I.; Suri, M., The hp-version of the finite element method with quasiuniform meshes, Model. math. anal. numer., 21, 199-238, (1987) · Zbl 0623.65113
[5]	Babuška, I.; Szabo, B.; Katz, I.N., The p-version of the finite element method, SIAM J. numer. anal., 18, 515-545, (1981) · Zbl 0487.65059
[6]	S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Users Manual, Tech. Report ANL-95/11, Argonne National Laboratory, 2004.
[7]	Bauer, A.C.; Patra, A.K., Performance of parallel preconditioners for adaptive hp-FEM discretizations of incompressible flows, Commun. numer. meth. eng., 18, 305-313, (2002) · Zbl 1005.76054
[8]	Bauer, A.C.; Patra, A.K., Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximation for linear elasticity with and without discontinuous coefficients, Int. J. numer. meth. eng., 59, 337-364, (2004) · Zbl 1047.74052
[9]	Davis, T.A., Algorithm 832: UMFPACK - an unsymmetric-pattern multifrontal method with a column pre-ordering strategy, ACM trans. math. software, 30, 196-199, (2004) · Zbl 1072.65037
[10]	I. Doležel, P. Šolín, M. Zítka, On the hp-FEM for singular electrostatics problems, in: A. Krawczyk, S. Wiak, X.M. Lopez-Fernendez (Eds.), Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering. Proceedings of ISEF ’05, IOS Press, Amsterdam, 2006, pp. 6.
[11]	M.A. Heroux, R.A. Bartlett, V.E. Howle, R.J. Hoekstra, J.J. Hu, T.G. Kolda, R.B. Lehoucq, K.R. Long, R.P. Pawlowski, E.T. Phipps, A.G. Salinger, H.K. Thornquist, R.S. Tuminaro, J.M. Willenbring, A. Williams, K.S. Stanley, An overview of the trilinos project, ACM Trans. Math. Software 31 (3) (2005) 397-423. · Zbl 1136.65354
[12]	S. Iqbal, G.F. Carey, Performance analysis of dynamic load balancing algorithms with variable number of processors, J. Parallel Distrib. Comput. 65 (8) (2005) 934-948. · Zbl 1072.68512
[13]	Karniadakis, G.E.; Sherwin, S.J., Spectral/hp element methods for CFD, (1999), Oxford University Press Oxford · Zbl 0954.76001
[14]	Laszloffy, A.; Long, J.; Patra, A.K., Simple data management, scheduling and solution strategies for managing the irregularities in parallel adaptive hp finite element simulations, Parallel comput., 26, 1765-1788, (2000) · Zbl 0948.68013
[15]	Melenk, J.M., hp-finite element methods for singular perturbations, (2002), Springer-Verlag Berlin · Zbl 1021.65055
[16]	Monk, P., Finite element methods for maxwell’s equations, (2003), Oxford University Press New York · Zbl 1024.78009
[17]	M. Paszynski, J. Kurtz, L. Demkowicz, Parallel, fully automatic hp-adaptive 2D finite element package, TICAM Report 04-07, The University of Texas at Austin, 2004. · Zbl 1093.65113
[18]	Patra, A.K., Parallel adaptive numerical simulation of dry avalanches over natural terrain, J. volcan. geotherm. res., 139, 1-21, (2005)
[19]	W. Rachowicz, D. Pardo, L. Demkowicz, Fully automatic hp-adaptivity in three dimensions, ICES Report 04-22, The University of Texas at Austin, 2004. · Zbl 1193.65203
[20]	Schwab, C.H., p- and hp-finite element methods, (1998), Clarendon Press Oxford
[21]	Shewchuk, J.R., Delaunay refinement algorithms for triangular mesh generation, Comput. geom., 22, 21-74, (2002) · Zbl 1016.68139
[22]	Šolín, P., Partial differential equations and the finite element method, (2005), John Wiley & Sons New Jersey
[23]	Šolín, P.; Demkowicz, L., Goal-oriented hp-adaptivity for elliptic problems, Comput. methods appl. mech. eng., 193, 449-468, (2004) · Zbl 1044.65082
[24]	Šolín, P.; Segeth, K.; Doležel, I., Higher-order finite element methods, (2004), Chapman & Hall/CRC Boca Raton