Modular hp-fem System HERMES and Its Application to the Maxwell s Equations 1

Transcription

1 Modular hp-fem System HERMES and Its Application to the Maxwell s Equations 1 Tomáš Vejchodský a Pavel Šolín b Martin Zítka b a Mathematical Institute, Academy of Sciences, Žitná 25, Praha 1, Czech Republic b Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas , USA Abstract In this paper we introduce the high-performance modular finite element system HERMES, a multi-physics hp-fem solver 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. Key words: hp-fem, time-harmonic Maxwell s equations, hierarchic higher-order edge elements 2000 MSC classification: 65N30 addresses: vejchod@math.cas.cz (Tomáš Vejchodský), solin@utep.edu (Pavel Šolín), zitka@math.utep.edu (Martin Zítka). 1 This work was supported by the Grant Agency of the Czech Republic, grants No. 201/04/P021 and No. 102/05/0629, and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z Preprint submitted to Elsevier Science 5 September 2005

2 1 Introduction The hp-fem is a sophisticated version of the finite element method (FEM) which varies both the diameter and polynomial degree of elements in order to maximize the convergence rates. Its advantage over other numerical methods is an unconditional exponential convergence, even for problems with singular solutions. The method was first introduced in the 1980s [4,5,2,3], and its theoretical foundations are well established today. However, the understanding of practical aspects of the method (such as the design of optimal data structures and algorithms, error estimates, automatic hp-adaptivity, etc.), has not yet attained a sufficient level of maturity. This can be illustrated on the fact that the hp-fem has not yet become a standard in commercial engineering codes. Nowadays there are several groups both in the U.S. [7,8,11,13,14,17 19,24,23] and in Europe [1,10,15,20] dealing with the hp-fem. In all cases, the investigation of the practical aspects is a substantial part of their research effort. 2 The hp-fem system HERMES The multi-physics modular finite element system HERMES has been developed by the authors over the last several years. The system is based on a novel modular approach where the finite element technology is completely separated from the physics of the problems represented by concrete PDEs. The modular structure of the system is illustrated in Fig. 1. FEM/hp FEM Module Continous Problem Geometry PDE Bdy. Cond. Mesh Generator Weak Formulation Edge elements Continuous elements Taylor Hood elements FEM/hp FEM Kernel Other types of elements Output of Results Visualization, etc. Discrete Problem smatrix Interface Solution stiffness matrix load vector Trilinos PETSc UMFPACK Other Algebraic Module solution coefficient vector Fig. 1. The modular structure of HERMES. 2

3 The heart of the solver is the FEM/hp-FEM Module which contains all PDEindependent algorithms, such as processing and adaptation of finite element meshes, numerical quadrature, assembling procedures, solution of systems of linear and nonlinear algebraic equations, a-posteriori error estimation, etc. The physics of the problem is represented via smaller PDE modules which are independent of the FEM/hp-FEM Module. The PDE modules contain not only the definition of concrete PDEs along with related physical parameters, but also scalar or vector-valued finite elements which are appropriate for their discretization. The PDE modules are strictly separated from the FEM/hp- FEM Module at each level of the hierarchic data structures. Last we would like to mention the module smatrix which serves as a universal interface between the FEM/hp-FEM Module and the Algebraic Module. The latter contains solver packages for sparse systems of linear and nonlinear algebraic equations, such as Trilinos [12], PETSc [6], and UMFPACK [9]. When these modules are not installed on the system where HERMES is run, smatrix can use a set of standard ILU-preconditioned matrix solvers. It is not technically difficult to add new solvers to the Algebraic Module. 3 Application to time-harmonic Maxwell s equations In addition to a module for systems of nonlinear elliptic equations and a module for the incompressible Navier Stokes equations, HERMES contains a time-harmonic Maxwell s equations module. Let us describe this module in more detail. Consider a bounded polygonal domain Ω R 2 whose boundary Ω is split into two relatively open disjoint parts Γ P and Γ I representing standard perfect conducting and impedance boundary conditions, respectively. The classical (strong) formulation of the Maxwell s equations reads: curl ( µ 1 r curl E ) κ 2 ɛ r E = F in Ω, E τ = 0 on Γ P, (1) µ 1 r curl E iκλe τ = g τ on Γ I. Here curl c = ( c/ x 2, c/ x 1 ) and curl b = b 2 / x 1 b 1 / x 2 are the standard vector and scalar curl operators, τ = ( ν 2, ν 1 ) is the positivelyoriented unit tangent vector to Ω, ν = (ν 1, ν 2 ) is the unit outer normal vector to Ω, and the symbol i stands for the imaginary unit. Some related notations are presented in Table 1. 3

4 E = E(x) C 2 phasor of the electric field strength (unknown) µ r = µ r (x) R relative permeability ɛ r = ɛ r (x) C 2 2 relative permittivity κ = const. R wave number λ = λ(x) > 0 impedance F = F(x) C 2 right-hand side of the PDE g = g(x) C 2 right-hand side of the impedance boundary condition Table 1 Quantities used in the time-harmonic Maxwell s equations. Problem (1) is formulated in the weak sense as follows: Find E V = {E H(curl, Ω) : E τ = 0 on Γ P } such that a(e, Φ) = F(Φ) Φ V, where the sesquilinear form a and the antilinear functional F are given by a(e, Φ) = F(Φ) = Ω Ω µ 1 r curl E curl Φ dx κ 2 F Φ dx + (g τ)(φ τ) ds. Γ I Ω (ɛ r E) Φ dx iκ Γ I λ(e τ)(φ τ) ds, We assume a triangulation T h,p of the domain Ω and assign a polynomial degree p j 0 to every element K j T h,p. This hp-mesh defines the following piecewise polynomial subspace of V : V h,p = { E h,p V : E h,p Kj P p j (K j ) and the tangent componet E h,p τ k is continuous on each edge e k E h,p }, where E h,p is the set of all edges in T h,p, τ k is the tangent vector to the edge e k and P p j (K j ) denotes the space of vector valued polynomials defined on a triangle K j with both components being of degree p j. With the finite element space V h,p problem: Find E h,p V h,p such that in hand, we can formulate the discrete a(e h,p, Φ h,p ) = F(Φ h,p ) Φ h,p V h,p. (2) The solution of this system of linear algebraic equations yields the approximate solution E h,p. For more details on the hp-fem including edge elements we refer to [24,22]. 4

5 4 Numerical experiments Let us use HERMES to illustrate the superior performance of the hp-fem over the standard low-order FEM. Consider the time-harmonic Maxwell s equations in the L-shape domain Ω with the vertices B 1 = (0, 0), B 2 = (0, 1), B 3 = (1, 1), B 4 = (1, 1), B 5 = ( 1, 1), and B 6 = ( 1, 0). Perfect conducting boundary conditions are prescribed on the edges meeting at the origin (B 1 B 2 and B 6 B 1 ) and the rest of Ω is equipped with impedance boundary conditions. For simplicity, the equation coefficients are chosen constant: µ r = 1, ɛ r = I (2 2 identity matrix), κ = 1, λ = 1. The exact solution is given by E = u, where u = r 2 3 sin ((2θ + π)/3). Hence, E 1 = 2 3 r 1 3 cos ( π 6 + θ 3 ), E 2 = 2 3 r 1 3 sin ( π 6 + θ 3 The solution is shown in the left part of Fig. 2. Notice the singularity at the reentrant corner (truncated for visualization purposes). ). Fig. 2. Example 1: Left: Modulus of the electric field phasor E. Right: Geometry of the mesh. The right hand sides F and g τ are chosen to be compatible with the exact solution: F = E, and g τ = ie τ on Ω. The problem was solved by the lowest-order FEM (p = 0 in all elements), piecewise-linear FEM (p = 1 in all elements), and by the hp-fem. We used the mesh shown in Fig. 2. In the first two cases we had to refine this meshes uniformly in order to obtain an accuracy comparable to the hp-fem: Every element in the mesh was subdivided into 10,000 triangles in the case p = 0 and into 484 triangles in the case p = 1. The distribution of the polynomial degree in the hp-mesh is shown in Fig. 3. 5

6 Fig. 3. Example 1: Meshes used for the hp-fem with detailed views of the re-entrant corners. The polynomial degree varies from 2 (blue) to 4 (red) in the left part and from 2 (blue) to 7 (red) in the right part. Table 2 compares the lowest-order case (p = 0) with the hp-fem on the relative error level of approx %. One can see that the hp-fem performed approx times faster compared to the lowest-order method. The memory requirements of the hp-fem were about times less. DOFs CPU time rel. error p = min 26 s % hp s % Ratio Table 2 Example 1: performance of the lowest-order and hp-fem. The other experiment compares the hp-fem (see the right part of Fig. 3) to the first-order method (p = 1). The corresponding results are presented in Table 3. In this case, the hp-fem was about 100 times faster and used 50 times less computer memory. DOFs CPU time rel. error p = min 18 s % hp s % Ratio Table 3 Example 1: performance of the first-order and hp-fem. 5 Summary and outlook We presented a multi-physics modular hp-fem system HERMES based on a novel approach, where the finite element technology is fully separated from the physics of the solved problems. In this way, new techniques implemented in the FEM/hp-FEM kernel work automatically for various PDEs, such as systems of linear and nonlinear second-order elliptic equations, time-harmonic 6

7 Maxwell s equations, convection-diffusion equations, Stokes problem, incompressible Navier Stokes equations, etc. The modular structure cuts down the development cost dramatically, and it greatly facilitates team-work. Due to its modular structure, HERMES allows for performing various numerical experiments (such as testing various sets of shape functions, various a-posteriori error estimators, various hp-adaptive strategies) very easily. The development of HERMES is by no means finished. Currently the implementation of hierarchic higher-order Taylor Hood elements is in progress, as well as an implementation of a novel approach to constrained approximation with multiple constraint levels, automatic hp-adaptivity, parallelization of the code, and other tasks. 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, [2] I. Babuška, W. Gui: 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 (1986), [3] I. Babuška, W. Gui: 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 (1986), [4] I. Babuška, M. Suri: The hp-version of the finite element method with quasiuniform meshes, Model. Math. Anal. Numer. 21 (1987), [5] I. Babuška, B. Szabo, I.N. Katz: The p-version of the finite element method, SIAM J. Numer. Anal. 18 (1981), [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, [7] A.C. Bauer, A.K. Patra: Performance of parallel preconditioners for adaptive hp-fem discretizations of incompressible flows, Commun. Numer. Meth. Engrg. 18 (2002), [8] A.C. Bauer, A.K. Patra: Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximation for linear elasticity with and without discontinuous coefficients, Int. J. Numer. Meth. Engrg. 59 (2004),

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