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Neumann–Neumann–Schur complement methods for Fekete spectral elements

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Abstract

For the iterative solution of the Schur complement system associated with the discretization of an elliptic problem by means of a triangular spectral element method (TSEM), Neumann–Neumann (NN) type preconditioners are constructed and studied. The TSEM approximation, based on Fekete nodes, is a generalization to non-tensorial elements of the classical Gauss–Lobatto–Legendre quadrilateral spectral elements. Numerical experiments show that the TSEM Schur complement condition number grows linearly with the polynomial approximation degree, N, and quadratically with the inverse of the mesh size, h. NN preconditioners for the Schur complement allow to reduce the N-dependence of the condition number, by solving local Neumann problems on each spectral element, and to eliminate the h-dependence if an additional coarse solver is employed. Numerical results indicate that, in spite of the more severe ill-conditioning, the condition number of the TSEM preconditioned operator satisfies the same bound as that of the standard SEM, i.e., Ch −2(1 + log N)2 for one-level NN preconditioning and C(1 + log N)2 for two-level Balancing Neumann-Neumann (BNN) preconditioning.

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References

  1. Schwab C (1998) p- and hp-Finite Element Methods Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, New York p 374

    Google Scholar 

  2. Karniadakis GE, Sherwin SJ (1999) Spectral hp Element Methods for CFD. Oxford Univ. Press, London p 390

    MATH  Google Scholar 

  3. Deville MO, Fischer PF, Mund EH (2002) High-order methods for incompressible fluid flow. Cambridge University Press, p 499

  4. Taylor MA, Wingate BA (1998) A generalized diagonal mass matrix spectral element method for non-quadrilateral elements. Appl Num Math 33:259–265

    Article  MathSciNet  Google Scholar 

  5. Hesthaven JS, Teng CH (2000) Stable spectral methods on tetrahedral elements. SIAM J Sci Comput 21:2352–2380

    Article  MATH  MathSciNet  Google Scholar 

  6. Warburton TA, Pavarino LF, Hesthaven JS (2000) A pseudo-spectral scheme for the incompressible Navier-Stokes equations using unstructured nodal elements. J Comput Phys 164:1–21

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Hesthaven JS, Warburton T (2002) Nodal high-order methods on unstructured grids. J Comp Phys 181:186–221

    Article  MATH  MathSciNet  Google Scholar 

  8. Giraldo FX, Warburton T (2005) A nodal triangle-based spectral element method for the shallow water equations on the sphere. J Comp Phys 207:129–150

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Chen Q, Babuvska I (1995) Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Comput Methods Appl Mech Engg 128:485–494

    MathSciNet  Google Scholar 

  10. Chen Q, Babuvska I (1996) The optimal symmetrical points for polynomial interpolation of real functions in a tetrahedron. Comput Methods Appl Mech Engg. 137:89–94

    Article  MATH  Google Scholar 

  11. Hesthaven JS (1998) From electrostatic to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J Numer Anal 35:655–676

    Article  MATH  MathSciNet  Google Scholar 

  12. Taylor MA, Wingate BA, Vincent RE (2000) An algorithm for computing Fekete points in the triangle. SIAM J Numer Anal 38:1707–1720

    Article  MATH  MathSciNet  Google Scholar 

  13. Bos L, Taylor MA, Wingate BA (2001) Tensor product Gauss–Lobatto points are Fekete points for the cube. Math Comput 70:1543–1547

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Pasquetti R, Rapetti F (2004) Spectral element methods on triangles and quadrilaterals: comparisons and applications. J Comput Phys 198:349–362

    Article  MATH  ADS  Google Scholar 

  15. Pasquetti R, Rapetti F (2006) Spectral element methods on unstructured meshes: comparisons and recent advances. J Sci Comput 27:377–387, (Proc. of the ICOSAHOM 2004 Congress)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pasquetti R, Pavarino LF, Rapetti F, Zampieri E (2006) Overlapping Schwarz preconditioners for Fekete spectral elements. In: Proc. of the 16th int. conf. on domain decomposition methods, Lecture Notes in Computational Science and Engineering

  17. Casarin MA (1997) Quasi-optimal Schwarz methods for the conforming spectral element discretization. SIAM J Numer Anal 34:2482–2502

    Article  MATH  MathSciNet  Google Scholar 

  18. Klawonn A, Pavarino LF (1998) Overlapping Schwarz methods for mixed linear elasticity and Stokes problems. Comput Meths Appl Mech Engg 165:233–245

    Article  MATH  MathSciNet  Google Scholar 

  19. Fischer PF (1997) An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J Comput Phys 133:84–101

    Article  MATH  MathSciNet  Google Scholar 

  20. Fischer PF, Lottes JW (2005) Hybrid Schwarz-Multigrid methods for the spectral element method: extensions to Navier–Stokes. J Sci Comput 6:345–390

    MathSciNet  Google Scholar 

  21. Toselli A, Widlund O (2005) Domain Decomposition Methods – Algorithms and Theory. Springer, Berlin Heidelberg New York, p 450

    MATH  Google Scholar 

  22. Pavarino LF (1997) Neumann–Neumann algorithms for spectral elements in three dimensions. RAIRO M^2AN 31: 471–493

    MATH  MathSciNet  Google Scholar 

  23. Pavarino LF, Widlund OB (2002) Balancing Neumann–Neumann methods for incompressible Stokes equations. Comm Pure Appl Math 55:302–335

    Article  MATH  MathSciNet  Google Scholar 

  24. Girault V, Raviart P-A (1986) Finite Element Methods for Navier–Stokes equations. Theory and Algorithms. Springer-Verlag, Berlin, p 374

    MATH  Google Scholar 

  25. Dubiner M (1991) Spectral methods on triangles and other domains. J Sci Comput 6:345–390

    Article  MATH  MathSciNet  Google Scholar 

  26. Bos L (1991) On certain configurations of points in \(\mathbb R^{n}\) which are unisolvent for polynomial interpolation. J Approx Theory 64:271–280

    Article  MATH  MathSciNet  Google Scholar 

  27. Stroud AH (1971) Approximate Calculations of Multiple Integrals. Prentice-Hall, p 431

  28. Cools R (2002) Advances in multidimensional integration. J Comput Appl Math 149:1–12

    Article  MATH  MathSciNet  Google Scholar 

  29. Demkowicz L, Walsh T, Gerdes K, Bajer A (1998) 2D hp-adaptative finite element package Fortran 90 implementation (2Dhp90). TICAM Report 98–14

  30. Stroud AH, Secrest D (1966) Gaussian Quadrature Formulas. Prentice-Hall, p 374

  31. Hu NH, Guo XZ, Katz IN (1998) Bounds for eigenvalues and condition numbers in the p-version of the finite element method. Math Comput 67:1423–1450

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Melenk JM (2002) On condition numbers in hp-FEM with Gauss-Lobatto based shape functions. J Comp Appl Math 139:21–48

    Article  MATH  MathSciNet  Google Scholar 

  33. Smith BF, Bjorstad PE, Gropp WD (1996) Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press p 224

  34. Mandel J, Brezina M (1996) Balancing domain decomposition for problems with large jumps in coefficients. Math Comp 65:1387–1401

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire J.-A. Dieudonné, CNRS & Université de Nice et Sophia-Antipolis, Parc Valrose, 06108, Nice cedex 02, France

    Richard Pasquetti & Francesca Rapetti

  2. Department of Mathematics, Universitá di Milano, Via Saldini 50, 20133, Milano, Italy

    Luca Pavarino & Elena Zampieri

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  1. Richard Pasquetti
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  2. Francesca Rapetti
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  3. Luca Pavarino
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  4. Elena Zampieri
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Correspondence to Richard Pasquetti.

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Pasquetti, R., Rapetti, F., Pavarino, L. et al. Neumann–Neumann–Schur complement methods for Fekete spectral elements. J Eng Math 56, 323–335 (2006). https://doi.org/10.1007/s10665-006-9066-x

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