## A parallel space-time boundary element method for the heat equation.(English)Zbl 1443.65201

Summary: In this paper we introduce a new parallel solver for the weakly singular space-time boundary integral equation for the heat equation. The space-time boundary mesh is decomposed into a given number of submeshes. Pairs of the submeshes represent dense blocks in the system matrices, which are distributed among computational nodes by an algorithm based on a cyclic decomposition of complete graphs ensuring load balance. In addition, we employ vectorization and threading in shared memory to ensure intra-node efficiency. We present scalability experiments on different CPU architectures to evaluate the performance of the proposed parallelization techniques. All levels of parallelism allow us to tackle large problems and lead to an almost optimal speedup.

### MSC:

 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations

Vc; BEM4I
Full Text:

### References:

 [1] Noon, P., The single layer heat potential and Galerkin boundary element methods for the heat equation (1988), University of Maryland, Thesis [2] Arnold, D. N.; Noon, P. J., Coercivity of the single layer heat potential, J. Comput. Math., 7, 2, 100-104 (1989), URL http://www.jstor.org/stable/43692419 · Zbl 0672.65092 [3] Costabel, M., Boundary integral operators for the heat equation, Integral Equations Operator Theory, 13, 498-552 (1990) · Zbl 0715.35032 [4] Hsiao, G. C.; Saranen, J., Boundary integral solution of the two-dimensional heat equation, Math. Methods Appl. Sci., 16, 2, 87-114 (1993) · Zbl 0772.45001 [5] Costabel, M., Time-dependent problems with the boundary integral equation method, (Stein, E.; de Borst, R.; Hughes, T. J.R., Encyclopedia of Computational Mechanics (2004), John Wiley & Sons), 703-721 [6] Lubich, C.; Schneider, R., Time discretization of parabolic boundary integral equations, Numer. Math., 63, 4, 455-481 (1992) · Zbl 0795.65061 [7] Chapko, R.; Kress, R., Rothe’s method for the heat equation and boundary integral equations, J. Integral Equations Appl., 9, 1, 47-69 (1997) · Zbl 0885.65101 [8] Tausch, J., Nyström discretization of parabolic boundary integral equations, Appl. Numer. Math., 59, 11, 2843-2856 (2009) · Zbl 1176.65103 [9] Costabel, M.; Saranen, J., The spline collocation method for parabolic boundary integral equations on smooth curves, Numer. Math., 93, 3, 549-562 (2003) · Zbl 1015.65055 [10] Messner, M.; Schanz, M.; Tausch, J., A fast Galerkin method for parabolic space – time boundary integral equations, J. Comput. Phys., 258, 15-30 (2014) · Zbl 1349.65716 [12] Messner, M.; Schanz, M.; Tausch, J., An efficient Galerkin boundary element method for the transient heat equation, SIAM J. Sci. Comput., 37, 3, A1554-A1576 (2015) · Zbl 1433.65200 [13] Dongarra, J., The international exascale software project roadmap, Int. J. High Perform. Comput. Appl., 25, 1, 3-60 (2011) [14] Speck, R.; Ruprecht, D.; Emmett, M.; Minion, M.; Bolten, M.; Krause, R., A space – time parallel solver for the three-dimensional heat equation, (Parallel Computing: Accelerating Computational Science and Engineering (CSE). Parallel Computing: Accelerating Computational Science and Engineering (CSE), Advances in Parallel Computing, vol. 25 (2014), IOS Press), 263-272 [15] Dongarra, J.; Hittinger, J.; Bell, J.; Chacon, L.; Falgout, R.; Heroux, M.; Hovland, P.; Ng, E.; Webster, C.; Wild, S., Applied mathematics research for exascale computing (2014), Department of Energy: Department of Energy US [16] Lions, J.-L.; Maday, Y.; Turinici, G., Résolution d’edp par un schéma en temps ¡¡ pararéel ¿¿, C. R. Acad. Sci., Paris I, 332, 7, 661-668 (2001) · Zbl 0984.65085 [17] Gander, M.; Neumüller, M., Analysis of a new space – time parallel multigrid algorithm for parabolic problems, SIAM J. Sci. Comput., 38, 4, A2173-A2208 (2016) · Zbl 1342.65225 [18] Lukas, D.; Kovar, P.; Kovarova, T.; Merta, M., A parallel fast boundary element method using cyclic graph decompositions, Numer. Algorithms, 70, 4, 807-824 (2015) · Zbl 1332.65177 [19] Kravcenko, M.; Merta, M.; Zapletal, J., Distributed fast boundary element methods for Helmholtz problems (2018), submitted for publication [20] Veit, A.; Merta, M.; Zapletal, J.; Lukáš, D., Efficient solution of time-domain boundary integral equations arising in sound-hard scattering, Internat. J. Numer. Methods Engrg., 107, 5, 430-449 (2016) · Zbl 1352.76080 [21] Kretz, M.; Lindenstruth, V., Vc: A C++ library for explicit vectorization, Softw. - Pract. Exp., 42, 11, 1409-1430 (2012) [22] Vector mathematical functions (2018), URL https://software.intel.com/en-us/mkl-developerreference-c-vector-mathematical-functions. Online (Accessed 29 August 2018) [23] OpenMP application programming interface (2015), URL https://www.openmp.org/wp-content/uploads/openmp-4.5.pdf. Online (Accessed 29 August 2018) [24] Arnold, D. N.; Noon, P. J., Boundary integral equations of the first kind for the heat equation, (Boundary elements IX, Vol. 3 (Stuttgart, 1987). Boundary elements IX, Vol. 3 (Stuttgart, 1987), Comput. Mech. (1987), Southampton), 213-229 [25] Lions, J. L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, vol. II (1972), Springer: Springer Berlin-Heidelberg-New York · Zbl 0227.35001 [26] Sauter, S.; Schwab, C., Boundary Element Methods (2011), Springer: Springer Berlin-Heidelberg [27] Hsiao, G. C.; Kopp, P.; Wendland, W. L., A Galerkin collocation method for some integral equations of the first kind, Computing, 25, 2, 89-130 (1980) · Zbl 0419.65088 [28] Sgallari, F., A weak formulation of boundary integral equations for time dependent parabolic problems, Appl. Math. Model., 9, 4, 295-301 (1985) · Zbl 0609.65080 [29] Radon, J., Zur mechanischen Kubatur, Monatsh. Math., 52, 4, 286-300 (1948) · Zbl 0031.31504 [30] Xeon platinum 8160 - Intel (2017), URL https://en.wikichip.org/wiki/intel/xeon_platinum/8160. Online (Accessed 26 September 2018)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.