Research Article
Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximations of linear elasticity with and without discontinuous coefficients
Andrew C. Bauer,
Abani K. Patra,
Andrew C. Bauer
Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, U.S.A.
Search for more papers by this authorCorresponding Author
Abani K. Patra
Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, U.S.A.
Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, U.S.A.Search for more papers by this authorAndrew C. Bauer,
Abani K. Patra,
Andrew C. Bauer
Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, U.S.A.
Search for more papers by this authorCorresponding Author
Abani K. Patra
Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, U.S.A.
Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, U.S.A.Search for more papers by this authorAbstract
Adaptive finite element methods (FEM) generate linear equation systems that require dynamic and irregular patterns of storage, access, and computation, making their parallelization difficult. Additional difficulties are generated for problems in which the coefficients of the governing partial differential equations have large discontinuities. We describe in this paper the development of a set of iterative substructuring based solvers and domain decomposition preconditioners with an algebraic coarse-grid component that address these difficulties for adaptive hp approximations of linear elasticity with both homogeneous and inhomogeneous material properties. Our solvers are robust and efficient and place no restrictions on the mesh or partitioning. Copyright © 2003 John Wiley & Sons, Ltd.
REFERENCES
- 1 Gui W, Babuška I. The h-, p- and hp-versions of the finite element method in one dimension. Part I: the error analysis of the p-version. Numerische Mathematik 1986; 49: 577–612.
- 2 Gui W, Babuška I. The h-, p- and hp-versions of the finite element method in one dimension. Part II: the error analysis of the h- and hp-versions. Numerische Mathematik 1986; 49: 613–657.
- 3 Gui W, Babuška I. The h-, p- and hp-versions of the finite element method in one dimension. Part III: the adaptive hp-version. Numerische Mathematik 1986; 49: 659–683.
- 4 Szabo B, Babuška I. Finite Element Analysis. Wiley: New York, 1991.
- 5 Balay S, Gropp W, McInnes LC, Smith B. PETSc 2.0 Users Manual. ANL-95/11 - Revision 2.0.28, Argonne National Laboratory, 2000.
- 6 Tuminaro RS, Heroux M, Hutchinson SA, Shadid JN. Official Aztec User's Guide Version 2.1. Technical Report SAND99-8801J, Sandia National Laboratory, 1999.
- 7 Laszloffy A, Long J, Patra A. Simple data management schemes and scheduling schemes for managing the irregularities in parallel adaptive hp finite element simulations. Parallel Computing 2000; 26: 1765–1788.
- 8 Patra A, Oden JT. Computational techniques for adaptive hp finite element methods. Finite Elements in Analysis and Design 1997; 25: 27–39.
- 9 Bauer AC, Sanjanwala S, Patra AK. Portable efficient solvers for adaptive finite element simulations of elastostatics in two and three dimensions. In Lecture Notes in Computational Science and Engineering: Vol. 23, Recent Developments in Domain Decomposition Methods, LF Pavarino, A Toselli (eds). Springer: Berlin, 2002; 223–243.
10.1007/978-3-642-56118-4_14 Google Scholar
- 10 Bramble JH, Pasciak JE, Schatz AH. The construction of preconditioners for elliptic problems by substructuring I. Mathematics of Computation 1986; 47(175): 103–134.
- 11 Widlund OB. Iterative substructuring methods: algorithms and theory for elliptic problems in the plane. In Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R Glowinski, GH Golub, GA Meurant, J Périaux (eds). SIAM: Philadelphia, PA, 1988; 113–128.
- 12 Dryja M. An additive schwarz algorithm for two- and three-dimensional finite element elliptic problems. In Domain Decomposition Methods, TF Chan, R Glowinski, J Periaux, OB Widlund (eds). SIAM: Philadelphia, PA, 1989; 168–172.
- 13 Mandel J. Iterative solvers by substructuring for the p-version finite element method. Computer Methods in Applied Mechanics and Engineering 1990; 80: 117–128.
- 14 Mandel J. Two level domain decomposition preconditioning for the p-version finite element method in three dimensions. International Journal for Numerical Methods in Engineering 1990; 29: 1095–1108.
- 15 Pavarino LF, Widlund OB. Iterative substructuring methods for spectral element discretizations of elliptic systems I. Compressible linear elasticity. SIAM Journal on Numerical Analysis 1999; 37(6): 353–374.
10.1137/S003614299732824X Google Scholar
- 16 Pavarino LF, Widlund OB. Iterative substructuring methods for spectral element discretizations of elliptic systems II. Mixed methods for linear elasticity and Stokes flow. SIAM Journal on Numerical Analysis 1999; 37(6): 375–402.
- 17 Ainsworth M. A preconditioner based on domain decomposition of h-p finite element approximation on quasi-uniform meshes. SIAM Journal on Numerical Analysis 1996; 33(4): 1358–1377.
- 18 Guo BQ, Cao W. Domain decomposition method for the h-p version finite element method. Computer Methods in Applied Mechanics and Engineering 1998; 157: 425–440.
- 19 Oden JT, Patra A, Feng YS. Domain decomposition solvers for adaptive hp finite element methods. SIAM Journal on Numerical Analysis 1997; 34(6): 2090–2118.
- 20 Mandel J. Balancing domain decomposition. Communications in Numerical Methods in Engineering 1993; 9: 233–241.
- 21 Smith B, Bjørstad P, Gropp W. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press: New York, 1996.
- 22 Sherwin SJ, Casarin M. Low energy basis preconditioning for elliptic substructured solvers based on unstructured spectral/ hp element discretisations. Journal of Computational Physics 2001; 171: 1–24.
- 23 Mandel J, Brezina M. Balancing domain decomposition for problems with large jumps in coefficients. Mathematics of Computation 1996; 65: 1387–1401.
- 24 Farhat C, Roux F. Implicit parallel processing in structural mechanics. In Computational Mechanics Advances, JT Oden (ed). North-Holland: Amsterdam, 1994; 1(1): 1–124.
- 25 Farhat C, Lesoinne M, LeTallec P, Pierson K, Rixen D. FETI-DP: a dual-primal unified FETI method—Part I: a faster alternative to the two-level FETI method. International Journal for Numerical Methods in Engineering 2001; 50(7): 1523–1544.
- 26 Bjørstad P, Dryja M. A coarse space formulation with good parallel properties for an additive Schwarz domain decomposition algorithm. Technical Report, http://www.ii.uib.no/petter/reports.html [15 October 2002].
- 27 Brandt A. Algebraic multigrid theory: the symmetric case. Applied Mathematics and Computation 1986; 19: 23–56.
- 28 Hackbush W. Multigrid Methods and Applications. Springer: Berlin, 1985.
10.1007/978-3-662-02427-0 Google Scholar
- 29 Mandel J. Étude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre. Comptes Rendus de l' Academie des Sciences de Paris Série I 1984; 298: 469–472.
- 30 Ruge JW, Stüben K. Algebraic multigrid. In Multigrid Methods, SF McCormick (ed), Frontiers in Applied Mathematics, Vol. 3. SIAM: Philadelphia, PA, 1987; 73–130.
10.1137/1.9781611971057.ch4 Google Scholar
- 31 Vaněk P. Acceleration of convergence of a two level algorithm by smoothing transfer operators. Applications of Mathematics 1992; 37: 265–274.
- 32 Vaněk P, Mandel J, Brezina M. Algebraic multigrid on unstructured meshes. UCD/CCM Report 34, Center for Computational Mathematics, University of Colorado at Denver, 1994.
- 33 Lasser C, Toselli A. Convergence of some two-level overlapping domain decomposition preconditioners with smoothed aggregation coarse spaces. In Lecture Notes in Computational Science and Engineering: 23, Recent Developments in Domain Decomposition Methods, LF Pavarino, A Toselli (eds). Springer: Berlin, 2002; 95–117.
- 34 Sarkis M. Partition of unity coarse spaces and Schwarz methods with harmonic overlap. In Lecture Notes in Computational Science and Engineering: 23, Recent Developments in Domain Decomposition Methods, LF Pavarino, A Toselli (eds). Springer: Berlin, 2002; 77–94.
- 35 Shames IH, Cozzarelli FA. Elastic and Inelastic Stress Analysis: Revised Printing. Taylor & Francis: Washington, DC, 1997.
10.1201/b16599 Google Scholar
- 36 Demkowicz L, Oden JT, Rachowicz W, Hardy O. Toward a universal h-p adaptive finite element strategy, Part I. Constrained approximation and data structure. Computer Methods in Applied Mechanics and Engineering 1989; 77: 9–112.
10.1016/0045-7825(89)90129-1 Google Scholar
- 37 Irons BM. A frontal solution program for finite element analysis. International Journal for Numerical Methods in Engineering 1970; 2: 5–32.
10.1002/nme.1620020104 Google Scholar
- 38 Benner RE, Montry GR, Weigand GG. Concurrent multifrontal methods: shared memory, cache, and frontwidth issues. International Journal of Supercomputer Applications 1987; 1: 26–44.
- 39 Zang WP, Liu EM. A parallel frontal solver on the Alliant. Computers and Structures 1991; 38: 202–215.
- 40 Duff IS, Scott JA. The use of multiple fronts in Gaussian elimination. Technical Report RAL 94-040, Rutherford Appleton Laboratory, 1994.
- 41 Luenberger DG. Introduction to Linear and Nonlinear Programming. Addison-Wesley: Reading, MA, 1973.
- 42 Golub GH, Van Loan CF. Matrix Computations ( 2nd edn). The Johns Hopkins University Press: Baltimore, 1989.
- 43 van de Geijn R, Gunnels J. PLAPACK Users Guide. http://www.cs.utexas.edu/users/plapack/ [21 October 2002].
- 44 Patra A, Laszloffy A, Long J. Data structures and load balancing for parallel adaptive hp finite element methods. Computers and Mathematics with Applications 2003; 46: 105–123.
- 45 Laszloffy A, Long J, Patra AK. Simple data management, scheduling and solution strategies for managing the irregularities in parallel adaptive hp finite element simulations. Parallel Computing 2000; 26: 1765–1788.
- 46 Long J. Integrated data management and dynamic load balancing for hp and generalized FEM. Ph.D. Dissertation, Mechanical and Aerospace Engineering Department, University at Buffalo, 2001.
- 47 Sagan H. Space Filling Curves. Springer: New York, 1994.
10.1007/978-1-4612-0871-6 Google Scholar
- 48 George A, Liu JWH. Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall: Englewood Cliffs, NJ, 1981.
- 49 Verfurth R. A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner Verlag and J. Wiley: Stuttgart, 1996.
