Abstract
In this paper we develop a technique for exploiting symmetry in the numerical treatment of boundary value problems (BVP) and eigenvalue problems which are invariant under a finite group \(\mathcal{G}\) of congruences of \({\rm{I\!R}}^{m}\). This technique will be based upon suitable restriction matrices strictly related to a system of irreducible matrix representation of \(\mathcal{G}\). Both Abelian and non-Abelian finite groups are considered. In the framework of symmetric Galerkin boundary element method (SGBEM), where the discretization matrices are typically full, to increase the computational gain we couple Panel Clustering Method [30] and Adaptive Cross Approximation algorithm [13] with restriction matrices introduced in this paper, showing some numerical examples. Applications of restriction matrices to SGBEM under the weaker assumption of partial geometrical symmetry, where the boundary has disconnected components, one of which is invariant, are proposed. The paper concludes with several numerical tests to demonstrate the effectiveness of the introduced technique in the numerical resolution of Dirichlet or Neumann invariant BVPs, in their differential or integral formulation.
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References
Aimi, A., Diligenti, M.: Stima dell’errore nel calcolo degli autovettori nel problema del “BUCKLING” di una piastra incastrata al bordo. Calcolo 30(2) 171–187 (1993)
Aimi, A., Diligenti, M., Monegato, G.: New numerical integration schemes for applications of galerkin BEM to 2D problems. Int. J. Numer. Methods Eng. 40 1977–1999 (1997)
Aimi, A., Diligenti, M.: Numerical integration in 3D Galerkin BEM solution of HBIEs. Comput. Mech. 28, 233–249 (2002)
Aimi, A., Diligenti, M., Lunardini, F., Salvadori, A.: A new application of the Panel Clustering Method for 3D SGBEM. CMES 4(1), 31–49 (2003)
Aimi, A., Bassotti, L., Diligenti, M.: Groups of congruences and restriction matrices. BIT 43(4), 671–693 (2003)
Aimi, A., Diligenti, M., Freddi, F., Salvadori, A.: Restriction matrices for SGBEM applications. Comput. Mech. 32, 430–444 (2003)
Aimi, A., Diligenti, M., Lunardini, F.: Panel clustering method and restriction matrices for symmetric Galerkin BEM. Numer. Algorithms 40, 355–382 (2005)
Allgower, E.L., Böhmer, K., Georg, K., Miranda, R.: Exploiting symmetry in boundary element methods. SIAM J. Numer. Anal. 29, 534–552 (1992)
Allgower, E.L., Georg, K., Miranda, R., Tausch, J.: Numerical exploitation of equivariance. Z. Angew. Math. Mech. 78, 795–806 (1998)
Allgower, E.L., Georg, K.: Numerical exploitation of symmetry in integral equations. Adv. Comput. Math. 9, 1–20 (1998)
Bassotti, L.: Linear operators that are T-invariant with respect to a group of homeomorphims. Russ. Math. Surv. 43(1), 67–101 (1988)
Bassotti, L.: Sulla molteplicitá degli autovalori di operatori lineari T-invarianti rispetto ad un gruppo di omeomorfismi. Rend. Accad. Naz. Sci. Detta dei XL 19, 35–45 (1989)
Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86, 565–589 (2000)
Bebendorf, M., Rjasanow, S.: Adaptive low-rank approximation of collocation matrices. Computing 70, 1–24 (2003)
Bonnet, M., Maier, G., Polizzotto, C.: Symmetric Galerkin boundary element method. Appl. Mech. Rev. 51, 669–704 (1998)
Bonnet, M.: Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations. Int. J. Numer. Methods Eng. 57, 1053–1083 (2003)
Bossavit, A.: Boundary value problems with symmetry and their approximation by finite elements. SIAM J. Appl. Math. 53(5), 1352–1380 (1993)
Bossavit, A.: On the computation of strains and stresses in symmetrical articulated structures. Lect. Appl. Math. 29, 111–123 (1993)
Brezzi, F., Douglas, C.C., Marini, L.D.: A parallel domain reduction method. Numer. Methods Partial Differ. Equ. 5, 195–202 (1989)
Costabel, M., Wendland, W.: Strong ellipticity of boundary integral operators. Crelle’s J. Reine Angew. Math. 372, 34–63 (1986)
Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)
Diligenti, M., Monegato, G.: Integral evaluation in the BEM solution of (hyper)singular integral equations. 2D problems on polygonal domains. J. Comput. Appl. Math. 81, 29–57 (1997)
Douglas, C.C., Mandel, J.: An abstract theory for the domain reduction method. Computing 48, 73–96 (1992)
Douglas, C.C., Smith, B.F.: Using symmetries and antisymmetries to analyze a parallel multigrid algorithm: the elliptic boundary value problem case. SIAM J. Numer. Anal. 26, 1439–1461 (1989)
Douglas, C.C.: Some nontelescoping parallel algorithms based on serial multigrid aggregation and disaggregation techniques, Multigrid Methods. Lect. Notes Pure Appl. Math. 110, 167–176 (1988)
Faessler, A., Stiefel, E.: Group Theoretical Methods and their Applications. Birkhauser Boston, Boston, MA (1992)
Giebermann, K.: Multilevel approximation of boundary integral operators. Computing 67(3), 183–207 (2001)
Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261, 1–21 (1997)
Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput Phys. 73(2), 325–348 (1987)
Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54, 463–491 (1989)
Hackbusch, W.: A sparse matrix arithmetic based on \(\cal H\)-matrices. Part I: Introduction to \({\cal H}\)-matrices. Computing 62(2), 89–109 (1999)
Hackbusch, W., Sauter, S.: On the efficient use of the Galerkin method to solve Fredholm integral equations. Appl. Math. 38, 301–322 (1993)
Hartmann, F.: Introduction to Boundary Elements. Springer, Berlin Heidelberg New York (1989)
Kurt, G., Miranda, R.: Symmetry aspects in numerical linear algebra with applications to boundary element methods. Lect. Appl. Math. 29, 213–228 (1993)
Kurz, S., Rain, O., Rjasanow, S.: Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems. Comput. Mech. 32, 423–429 (2003)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer, Berlin Heidelberg New York (1972)
Lobry, J., Broche, C.H.: Geometrical symmetry in the boundary element method. Eng. Anal. Bound. Elem. 14, 229–238 (1994)
Lutz, E., Ye, W., Mukherjee, S.: Elimination of rigid body modes from discretized boundary integral equations. Int. J. Solids Struct. 35(33), 4427–4436 (1998)
Nedelec, J.: Curved finite element method for the solution of singular integral equations on surface \({\rm{I\!R}}^{3}\). Comput. Methods Appl. Mech. Eng. 8, 61–80 (1976)
Sauter, S.A.: Variable order panel clustering. Computing 64, 223–261 (2000)
Schwab, C., Wendland, W.: Kernel properties and representations of boundary integral operators. Math. Comput. Methods Appl. Mech. Eng. 83, 69–89 (1990)
Serre, J.P.: Linear Representations of Finite Groups, Graduate Texts in Mathematics 42. Spinger Berlin Heidelberg, New York (1977)
Smirnov, V.I.: Linear Algebra and Group Theory. McGraw Hill, New York (1961)
Smith, G.F.: Projection operators for symmetric regions. Arch. Ration. Mech. Anal. 54, 161–174 (1974)
Tausch, J.: A generalization of the discrete Fourier transform. In: Allgower, E.L., Georg, K., Miranda, R. (eds.) Exploiting Symmetry in Applied and Numerical Analysis, Lectures in Applied Mathematics 29, pp. 405–412. AMS, Providence, RI (1993)
Verchery, G.: Régularisation du systéme de l’équilibre des structures élastiques discrètes. C.R. Acad. Sci. Paris 311(II), 585–589 (1990)
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Aimi, A., Diligenti, M. Restriction matrices for numerically exploiting symmetry. Adv Comput Math 28, 201–235 (2008). https://doi.org/10.1007/s10444-006-9019-y
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DOI: https://doi.org/10.1007/s10444-006-9019-y