# zbMATH — the first resource for mathematics

Domain decomposition and parallel processing of a finite element model of the shallow water equations. (English) Zbl 0784.76052
Summary: We present nonoverlapping domain decomposition techniques applied to a two-stage Numerov-Galerkin finite element model of the shallow water equations over a limited-area domain. The Schur complement matrix formulation is employed and a modified interface matrix approach is proposed to handle the coupling between subdomains. The resulting nonsymmetric Schur complement matrices, modified interface matrices as well as the subdomain coefficient matrices are solved using preconditioned conjugate gradient squared non-symmetric iterative solvers. Various stages of the finite element solution are parallelized and the code is implemented on a four processor CRAY Y-MP supercomputer applying multitasking techniques in a dedicated environment.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 65Y05 Parallel numerical computation
CGS; FEUDX
Full Text:
##### References:
 [1] Schwarz, H.A., Über einige abbildungsaufgaben, Ges. math. abh., 11, 65-83, (1869) [2] Przemienniechi, J.S., Matrix structural analysis of substructures, Aiaa j., 1, 138-147, (1963) [3] Kopriva, D.A., Computation of hyperbolic equations on complicated domains with patched and overset Chebyshev grids, SIAM J. sci. statist. comput., 10, 120-132, (1989) · Zbl 0724.65092 [4] Rai, M.M., A conservative treatment of zonal boundaries for Euler equation calculations, J. comput. phys., 62, 472-503, (1986) · Zbl 0619.65085 [5] Kopriva, D.A., Domain decomposition with both spectral and finite difference methods for the accurate computation of flows with shocks, Appl. numer. math., 6, 141-151, (1989) · Zbl 0678.76059 [6] Temam, R., Survey of the status of finite element methods for partial differential equations, (), 1-33 [7] Kopriva, D.A., Spectral solution of inviscid supersonic flows over wedges and axisymmetric cones, Comput. & fluids, 21, 2, 247-266, (1992) · Zbl 0753.76135 [8] Glowinski, R.; Kinton, W.; Wheeler, M.F., Acceleration of domain decomposition algorithms for mixed finite elements by multi-level methods, (), 263-289 · Zbl 0704.65081 [9] Dinh, Q.V.; Glowinski, R.; Périaux, J.; Terrasson, G., On the coupling of viscous and inviscid models for incompressible fluid flows via domain decomposition, (), 350-369 · Zbl 0652.76023 [10] Glowinski, R.; Périaux, J.; Terrasson, G., On the coupling of viscous and inviscid models for compressible fluid flows via domain decomposition, (), 64-97 · Zbl 0703.76055 [11] Concus, P.; Golub, G.H.; O’Leary, D.P., A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, (), 309-332 · Zbl 0385.65048 [12] Dryja, M., A capacitance matrix method for Dirichlet problem on polygon regions, Numer. math., 39, 51-64, (1982) · Zbl 0478.65062 [13] Dryja, M., A finite element-capacitance method for elliptic problems on regions partitioned into subregions, Numer. math., 44, 153-168, (1984) · Zbl 0568.65075 [14] Golub, G.H.; Mayers, D., The use of preconditioning over irregular regions, (), 3-14 [15] Bjørstad, P.E.; Widlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. numer. anal., 23, 1097-1120, (1986) · Zbl 0615.65113 [16] Chan, T.F., Analysis of preconditioners for domain decomposition, SIAM J. numer. anal., 24, 382-390, (1987) · Zbl 0625.65100 [17] Chan, T.F., A domain-decomposed fast Poisson solver on a rectangle, SIAM J. sci. statist. comput., 8, s14-s26, (1987) · Zbl 0624.65100 [18] Chan, T.F.; Resasco, D.C., A framework for the analysis and construction of domain decomposition preconditioners, (), 217-230 [19] Cullen, M.J.P.; Morton, K.W., Analysis of evolutionary error in finite-element and other methods, J. comput. phys., 34, 245-267, (1980) · Zbl 0477.65064 [20] Navon, I.M., A survey of finite-element methods in quasi-linear fluid flow problems, (), 44 [21] Navon, I.M., Finite-element simulations of the shallow-water equations model on a limited area domain, Appl. math. modelling, 3, 337-348, (1979) · Zbl 0438.76017 [22] Navon, I.M.; Muller, U., FESW - A finite-element fortan IV program for solving the shallow-water equations, Adv. engrg. software, 1, 77-86, (1979) · Zbl 0412.65055 [23] Navon, I.M.; Riphagen, H.A., An implicit compact fourth-order algorithm for solving the shallow-water equations in conservative law-form, Monthly weather rev., 107, 1107-1127, (1979) · Zbl 0434.65079 [24] Navon, I.M., A numerov-Galerkin technique applied to a finite-element shallow-water equations model with enforced conservation of integral invariants and selective lumping, J. comput. phys., 52, 313-339, (1983) · Zbl 0526.76032 [25] Navon, I.M., FEUDX: A two-stage, high-accuracy finite-element FORTRAN program for solving shallow-water equations, Comput. geosci., 13, 255-285, (1987) [26] Kawahara, M.; Hirano, H.; Tsubota, K.; Inagaki, K., Elective lumping finite-element method for shallow-water flow, Internat. J. numer. methods fluids, 2, 89-112, (1982) · Zbl 0483.76023 [27] Neta, B.; Williams, R.T., Stability and phase speed for various finite-element formulations of the advection equation, Comput. & fluids, 14, 393-410, (1986) · Zbl 0613.76099 [28] Steppler, J.; Navon, I.M.; Lu, H.-I., Finite-element schemes for extended integrations of atmospheric models, J. comput. phys., 89, 95-124, (1990) · Zbl 0696.76031 [29] Sonneveld, P., CGS, A fast Lanczos-solver for nonsymmetric linear systems, SIAM J. sci. statist. comput., 10, 36-52, (1989) · Zbl 0666.65029 [30] Chan, T.F.; Resasco, D., A survey of preconditioners for domain decomposition, () [31] Bramble, J.H.; Pasciak, J.E.; Schatz, A.H., An iterative method for elliptic problems on regions partitioned into substructures, Math. comp., 46, 361-369, (1986) · Zbl 0595.65111 [32] Bramble, J.H.; Pasciak, J.E.; Schatz, A.H., The construction of preconditioners for elliptic problems by substructures, Math. comp., 47, 103-134, (1986) · Zbl 0615.65112 [33] Chan, T.F.; Goovaerts, D., A note on the efficiency of domain decomposed incomplete factorizations, SIAM J. sci. statist. comput., 11, 794-803, (1990) · Zbl 0707.65083 [34] Börgers, C., The Neumann-Dirichlet domain decomposition method with inexact solvers on the subdomains, Numer. math., 55, 123-136, (1989) · Zbl 0652.65080 [35] Y. Cai and I.M. Navon, Parallel processing of a finite element model of the shallow water equations via domain decomposition - A domain-decomposed preconditioner approach, unpublished. · Zbl 0784.76052 [36] Grammeltvedt, A., A survey of finite difference scheme for the primitive equations for a barotropic fluid, Monthly weather rev., 97, 384-404, (1969) [37] Schwarz, B.K.; Wendroff, B., The relative efficiency of finite-difference and finite-element methods, I. hyperbolic problems and splines, SIAM J. numer. anal., 11, 979-993, (1974) · Zbl 0294.65055 [38] Von-Rosenberg, D.U., Methods for the numerical solution of partial differential eqations, (1969), Elsevier New York · Zbl 0194.18904 [39] Ahlberg, H.H.; Nielson, E.N.; Walsh, J.L., () [40] Zienkiewicz, O.C.; Morgan, K., Finite elements and approximation, (1983), Wiley New York · Zbl 0582.65068 [41] Rubinstein, M.F., Combined analysis by substructures and recursion, ASCE J. structural div., 93, 231-235, (1967) [42] Furnike, T., Computerized multiple level substructuring analysis, Comput. & structures, 2, 1063-1073, (1972) [43] Cottle, R., Manifestations of the Schur complement, Linear algebra appl., 8, 189-211, (1974) · Zbl 0284.15005 [44] Hockney, R.W., The potential calculation and some applications, Methods comput. phys., 9, 135-211, (1970) [45] Ortega, J.M., Introduction to parallel and vector solution of linear systems, (1988), Plenum New York · Zbl 0669.65017 [46] Saad, Y., Krylov subspace methods on supercomputers, SIAM J. sci. statist. comput., 10, 1200-1232, (1989) · Zbl 0693.65028 [47] Dongarra, J.J.; Duff, L.S.; Sorensen, D.C.; Van der Vorst, H.A., Solving linear systems on vector and shared memory computers, (1991), SIAM Philadelphia, PA [48] Chronopoulos, A.T., A fast squared Lanczos method for nonsymmetric linear systems, () · Zbl 0836.65045 [49] Radicati, G.; Robert, Y.; Succi, S., Iterative algorithms for the solution of nonsymmetric systems in the modelling of weak plasma turbulence, J. comput. phys., 80, 489-497, (1989) · Zbl 0659.76126 [50] Kaasschieter, E.F., The solution of non-symmetric linear systems by bi-conjugate gradients or conjugate gradient squared, () · Zbl 0659.65031 [51] Keyes, D.E.; Gropp, W.D., A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM J. sci. statist. comput., 8, s166-s202, (1987) · Zbl 0619.65088 [52] Meijerink, J.A.; van der Vorst, H.A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric $$M- matrix$$, Math. comp., 31, 148-162, (1977) · Zbl 0349.65020 [53] Gustafsson, I.; A, class of first order factorization methods, Bit, 18, 142-156, (1978) [54] Gustafsson, I., Modified incomplete Cholesky (MIC) methods, (), 265-294 · Zbl 0767.65017 [55] Axelsson, O.; Barker, V.A., Finite solutions of boundary value problems, () · Zbl 0981.65130 [56] Keyes, D.E.; Gropp, W.D., Domain decomposition for nonsymmetric systems of equations: examples from computational fluid dynamics, (), 321-339 [57] Hageman, L.A.; Young, D.M., Applied iterative methods, (1981), Academic Press New York · Zbl 0459.65014 [58] () [59] Natarajan, R., Finite element applications on a shared-memory multiprocessor: algorithms and experimental results, J. comput. phys., 94, 352-381, (1991) · Zbl 0717.76079 [60] Gropp, W.D.; Keyes, D.E., Complexity of parallel implementation of domain decomposition techniques for elliptic partial differential equations, SIAM J. sci. statist. comput., 9, 312-326, (1988) · Zbl 0645.65069 [61] Brawer, S., Introduction to parallel programming, (1989), Academic Press New York · Zbl 0669.68020
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.