×

Computational aspects of the local discontinuous Galerkin method on unstructured grids in three dimensions. (English) Zbl 1286.65154

Summary: Using tensor notation, a simplified description of the most relevant operators of the local discontinuous Galerkin (LDG) method applied to a general elliptic boundary value problem on unstructured meshes in three dimensions is presented. A reduction of storage is achieved by introducing a fast algorithm for the assembly of the Schur complement. A semi-algebraic multilevel preconditioner for low-order approximations using the classical Lagrange interpolatory basis is discussed. A series of numerical experiments is presented to illustrate the performance of the proposed preconditioning technique and accuracy of the method on three-dimensional problems.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cockburn, B.; Shu, C., The Local Discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118
[2] Castillo, P.; Cockburn, B.; Perugia, I.; Schötzau, D., An a priori error analysis of the Local Discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38, 1676-1706 (2000) · Zbl 0987.65111
[3] Perugia, I.; Schötzau, D., An \(h p\) analysis of the Local Discontinuous Galerkin method for diffusion problems, J. Sci. Comput., 17, 561-571 (2002) · Zbl 1001.76060
[4] Castillo, P., Performance of discontinuous Galerkin methods for elliptic PDE’s, SIAM J. Sci. Comput., 24, 524-547 (2002) · Zbl 1021.65054
[5] Bustinza, R.; Gatica, G., A Local Discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions, SIAM J. Sci. Comput., 26, 152-177 (2004) · Zbl 1079.65114
[6] Yan, J.; Shu, C.-W., A Local Discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40, 769-791 (2002) · Zbl 1021.65050
[7] Yan, J.; Shu, C.-W., Local Discontinuous Galerkin methods for partial differential equations with higher order derivatives, J. Sci. Comput., 17, 27-47 (2002) · Zbl 1003.65115
[8] Cockburn, B.; Kanschat, G.; Perugia, I.; Schötzau, D., Superconvergence of the Local Discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39, 264-285 (2001) · Zbl 1041.65080
[9] Bangerth, W.; Hartmann, R.; Kanschat, G., Deal.II: a general purpose object oriented finite element library, ACM. Trans. Math. Software, 33, 24:1-24:27 (2007) · Zbl 1365.65248
[10] Strouttrup, B., The C++ Programming Language (1991), Addison-Wesley: Addison-Wesley Reading, MA
[11] Castillo, P., Stencil reduction algorithms for the Local Discontinuous Galerkin method, Internat. J. Numer. Methods Engrg., 81, 1475-1491 (2010) · Zbl 1183.76801
[12] Castillo, P.; Koning, J.; Rieben, R.; White, D., A discrete differential forms framework for computational electromagnetism, Comput. Model. Eng. Sci., 5, 331-346 (2004) · Zbl 1109.78322
[13] Castillo, P.; Rieben, R.; White, D., FEMSTER: an object oriented class library of high-order discrete differential forms, ACM Trans. Math. Software, 31, 425-457 (2005) · Zbl 1136.78330
[14] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics USA · Zbl 1002.65042
[15] Chow, E.; Heroux, M., An object-oriented framework for block preconditioning, ACM Trans. Math. Software, 24, 159-183 (1998) · Zbl 0930.65052
[17] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method (2009), Dover Publications
[18] Kanschat, G., Preconditioning methods for Local Discontinuous Galerkin discretizations, SIAM J. Sci. Comput., 25, 815-831 (2003) · Zbl 1048.65110
[19] Gopalakrishnan, J.; Kanschat, G., A multilevel discontinuous Galerkin method, Numer. Math., 95, 527-550 (2003) · Zbl 1044.65084
[20] Castillo, P.; Velázquez, E., A numerical study of a semi-algebraic multilevel preconditioner for the Local Discontinuous Galerkin method, Internat. J. Numer. Methods Engrg., 79, 255-268 (2008) · Zbl 1159.76348
[21] Xu, J., The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56, 215-235 (1996) · Zbl 0857.65129
[23] Saad, Y., ILUM: a multi-elimination ILU preconditioner for general sparse matrices, SIAM J. Sci. Comput., 17, 830-847 (1996) · Zbl 0858.65029
[24] Saad, Y.; Zhang, J., BILUTM: a domain-based multi-level block ILUT preconditioner for general sparse matrices, SIAM J. Matrix Anal. Appl., 21, 279-299 (1998) · Zbl 0942.65045
[26] Ma, C.-C.; Chang, S.-W., Analytical exact solutions of heat conduction problems for anisotropic multi-layered media, Heat Mass Transfer, 47, 1643-1655 (2004) · Zbl 1057.80003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.