Abstract
In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincaré–Steklov operators (Dirichlet–Neumann mapping) for interior and exterior problems in the presence of a nested mesh refinement. Our approach is based on the multilevel interface solver applied to the Schur complement reduction onto the nested refined interface associated with a nonmatching decomposition of a polygon by rectangular substructures. This paper extends methods from Khoromskij and Prössdorf (1995), where the finite element approximations of interior problems on quasi‐uniform grids have been considered. For both interior and exterior problems, the matrix–vector multiplication with the compressed Schur complement matrix on the interface is shown to have a complexity of the order O(N r log3 N u), where Nr = O((1 + p r) N u) is the number of degrees of freedom on the polygonal boundary under consideration, N u is the boundary dimension of a finest quasi‐uniform level and p r ⩾ 0 defines the refinement depth. The corresponding memory needs are estimated by O(N r logq N u), where q = 2 or q = 3 in the case of interior and exterior problems, respectively.
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References
V.I. Agoshkov and V.I. Lebedev, Poincaré–Steklov operators and domain decomposition methods in variational problems, in: Vychisl. Protsessy and Sistemy, Vol. 2 (Nauka, Moscow, 1985) pp. 173–227 (in Russian).
B. Alpert, G. Beylkin, R.R. Coifman and V. Rokhlin, Wavelet-like bases for the fast solution of second-kind integral equations, SIAM J. Sci. Statist. Comput. 14(1) (1993) 159–189.
N.S. Bakhvalov and M. Yu. Orekhov, On fast methods for the solution of Poisson equation, Zh. Vychisl. Mat. Mat. Fiz. 22(6) (1982) 1386–1392 (in Russian).
G. Beylkin, R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. XLIV (1991) 141–183.
F. Bornemann and P. Deufelhard, The cascadic multigrid method for elliptic problems, Numer. Math. 75 (1996) 135–152.
J.H. Bramble and J.T. King, A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries, Math. Comp. 63(207) (1994) 1–17.
J.H. Bramble, J.E. Pasciak and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990) 1–22.
M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19(3) (1988) 613–625.
W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations I: Stability and convergence, Math. Z. 215 (1994) 583–620.
W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993) 259–335.
W. Dahmen, S. Prössdorf and R. Schneider, Multiscale methods for pseudo-differential equations, in: Recent Advances in Wavelet Analysis, eds. L.L. Schumaker and G. Webb, Wavelet Analysis and its Applications 3 (Academic Press, 1994) pp. 191–235.
W. Dahmen, B. Kleemann, S. Prössdorf and R. Schneider, A multiscale method for the double layer potential equation on a polyhedron, in: Advances in Computational Mathematics, eds. H.P. Dikshit and C.A. Micchelli (World Scientific, Singapore, 1994) pp. 15–57.
M. Dryja and O.B. Widlund, Multilevel additive methods for elliptic finite element problems, in: Parallel Algorithms for PDEs, Proc. 6th GAMM-Seminar, ed. W. Hackbusch, Kiel, January 19–21, 1990 (Vieweg, Braunschweig, 1991) pp. 58–69.
P. Grisvard, Boundary Value Problems in Non-Smooth Domains (Pitman, London, 1985).
W. Hackbusch and Z.P. Nowak, On the fast matrix multiplication in the boundary element method by panel clustering, Numer. Math. 54 (1989) 463–491.
G.C. Hsiao, B.N. Khoromskij and W.L. Wendland, Boundary integral operators and domain decomposition, Preprint Math. Inst. A, 94-11, University of Stuttgart (1994).
B.N. Khoromskij, On fast computations with the inverse to harmonic potential operators via domain decomposition, J. Numer. Linear Algebra Appl. 3(2) (1996) 91–111.
B.N. Khoromskij, G.E. Mazurkevich and E.P. Zhidkov, Domain decomposition methods for elliptic problems in unbounded domains, Preprint JINR, E11-91-487, Dubna (1991).
B.N. Khoromskij and S. Prössdorf, Multilevel preconditioning on the refined interface and optimal boundary solvers for the Laplace equation, Adv. Comput. Math. 4 (1995) 331–355.
B.N. Khoromskij and G. Schmidt, Fast interface solvers for biharmonic Dirichlet problem on polygonal domains, Preprint No 162, WIAS, Berlin (1995); Numer. Math. (1997, to appear).
B.N. Khoromskij and W.L. Wendland, Spectrally equivalent preconditioners for boundary equations in substructuring techniques, East–West J. Numer. Math. 1(1) (1992) 1–26.
B.N. Khoromskij and G. Wittum, An asymptotically optimal Schur complement reduction for the Stokes equation, Preprint No 96/3, ICA3, University of Stuttgart (1996).
B.N. Khoromskij and G. Wittum, An asymptotically optimal substructuring method for the Stokes equation, in: DD9 Proceedings, eds. P. Bjorstad, M. Espedal and D. Keyes (Wiley, New York, 1997).
J.L. Lions and E. Magenes, Noin-Homogeneous Boundary Value Problems and Applications I (Springer, New York, 1972).
P. Oswald, Stable subspace splitting for Sobolev spaces and their applications, Preprint 93/07, University of Jena (1993).
P. Oswald, Multilevel Finite Element Approximations: Theory and Applications (Teubner, Stuttgart, 1994).
T. von Petersdorff and C. Schwab, Wavelet approximations for first kind boundary integral equations on polygons, Numer. Math. 74 (1996) 479–516.
T. von Petersdorff, C. Schwab and R. Schneider, Multiwavelets for second kind integral equations, Techn. Note, Inst. for Phys. Sci. and Techn., University of Maryland at College Park (September 1994), SIAM J. Numer. Anal. (to appear).
A. Rathsfeld, A wavelet algorithm for the solution of the double layer potential equation over polygonal boundaries, J. Integral Equations Appl. 7 (1995) 47–98.
V.V. Shaidurov, Some estimates of the rate of convergence for the cascadic conjugate gradient method, Comp. Math. Appl. 31, no.415 (1996) 161–171.
V.V. Shaidurov and L. Tobiska, The convergence of the cascadic conjugate-gradient method applied to elliptic problems in domains with angles, Preprint University Magdeburg (1996).
E.P. Stephan and T. Tran, A multi-level additive Schwarz method for hypersingular integral equations, Preprint University of Hannover (August 1994).
W.L. Wendland, Strongly elliptic boundary integral equations, in: The State of the Art in Numerical Analysis, eds. A. Iserles and M. Powell (Clarendon Press, Oxford, 1987) pp. 511–561.
E.P. Zhidkov and B.N. Khoromskij, Numerical algorithms on a sequence of grids and their applications in magnetostatics and theoretical physic problems, in: Elementary Particles and Nuclear Physics 19, v. 3 (Energoatomizdat, Moscow, 1988) pp. 622–668 (in Russian).
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Khoromskij, B.N., Prössdorf, S. Fast computations with the harmonic Poincaré–Steklov operators on nested refined meshes. Advances in Computational Mathematics 8, 111–135 (1998). https://doi.org/10.1023/A:1018988028583
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DOI: https://doi.org/10.1023/A:1018988028583