Abstract
We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are non-local, the inverse matrices will in general be dense, therefore representing them by standard techniques will require prohibitively large amounts of storage. In the field of integral equations, a successful technique for handling dense matrices efficiently is to use a data-sparse representation like the popular multipole method. In this paper we prove that this approach can be generalized to cover inverse matrices corresponding to partial differential equations by switching to data-sparse \({\mathcal{H}}\)- and \({\mathcal{H}^2}\)-matrices. The key results are existence proofs for local low-rank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for \({\mathcal{H}}\)- and \({\mathcal{H}^2}\)-matrix approximations of the entire matrices.
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Bebendorf M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J. Numer. Anal. 45(4), 1472–1494 (2007)
Bebendorf M., Hackbusch W.: Existence of \({\mathcal{H}}\)-matrix approximants to the inverse FE-matrix of elliptic operators with L ∞-coefficients. Numer. Math. 95, 1–28 (2003)
Bernardi C., Girault V.: A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35(5), 1893–1916 (1998)
Börm S.: Data-sparse approximation of non-local operators by \({\mathcal{H}^2}\)-matrices. Linear Algebra Appl. 422, 380–403 (2007)
Börm S.: Construction of data-sparse \({\mathcal{H}^2}\)-matrices by hierarchical compression. SIAM J. Sci. Comput. 31(3), 1820–1839 (2009)
Börm, S., Grasedyck, L., Hackbusch, W.: Hierarchical matrices. Lecture Note 21 of the Max Planck Institute for Mathematics in the Sciences (2003)
Börm S., Hackbusch W.: Data-sparse approximation by adaptive \({\mathcal{H}^2}\)-matrices. Computing 69, 1–35 (2002)
Ciarlet, P.G.: The finite element method for elliptic problems. SIAM (2002)
Clément P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)
Dahmen W., Faermann B., Graham I.G., Hackbusch W., Sauter S.A.: Inverse inequalities on non-quasiuniform meshes and applications to the mortar element method. Math. Comp. 73, 1107–1138 (2004)
Graham I.G., Lechner P., Scheichl R.: Domain decomposition for multiscale PDEs. Numer. Math. 106, 589–626 (2007)
Grasedyck, L.: Theorie und Anwendungen Hierarchischer Matrizen. Doctoral thesis, Universität Kiel (2001)
Grasedyck L., Hackbusch W.: Construction and arithmetics of \({\mathcal{H}}\)-matrices. Computing 70, 295–334 (2003)
Grasedyck, L., Kriemann, R., LeBorne, S.: Parallel black box domain decomposition based \({\mathcal{H}}\)-LU preconditioning. Technical Report 115, Max Planck Institute for Mathematics in the Sciences, Leipzig (2005)
Hackbusch W.: Multi-Grid Methods and Applications. Springer, Berlin (1985)
Hackbusch W.: A sparse matrix arithmetic based on \({\mathcal{H}}\)-matrices. Part I: Introduction to \({\mathcal{H}}\)-matrices. Computing 62, 89–108 (1999)
Hackbusch W., Khoromskij B.N., Sauter S.A.: On \({\mathcal{H}^2}\)-matrices. In: Bungartz, H., Hoppe, R., Zenger, C. (eds) Lectures on Applied Mathematics, pp. 9–29. Springer, Berlin (2000)
Schenk O., Gärtner K., Fichtner W., Stricker A.: PARDISO: A high-performance serial and parallel sparse linear solver in semiconductor device simulation. J. Future Gener. Comput. Syst. 18, 69–78 (2001)
Scott L.R., Zhang S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)
Xu J., Zhu Y.: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Models Meth. Appl. Sci. 18(1), 77–105 (2008)
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Börm, S. Approximation of solution operators of elliptic partial differential equations by \({\mathcal{H}}\)- and \({\mathcal{H}^2}\)-matrices. Numer. Math. 115, 165–193 (2010). https://doi.org/10.1007/s00211-009-0278-7
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DOI: https://doi.org/10.1007/s00211-009-0278-7