# zbMATH — the first resource for mathematics

Parallel, adaptive finite element methods for conservation laws. (English) Zbl 0826.65084
The authors explore extensively how to numerically solve hyperbolic conservation laws by means of nonconforming finite elements; computations are to be carried on parallel computers and a significant set of examples in one and two dimensions is analyzed, with an explicit Runge-Kutta method being employed for time discretization. In this way, discontinuities that may develop with time in hyperbolic phenomena are implicit in the discretization from the onset. Adaptive $$h$$- and $$p$$- refinements are shown to provide accuracy at low computational cost, when compared with fixed meshes computations. The paper sets the lines for further developments combining both types of refinement, in order “to optimize computational work in both smooth and discontinuous solution regions” (sic).

##### MSC:
 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65Y05 Parallel numerical computation 35L65 Hyperbolic conservation laws
Full Text:
##### References:
 [1] S. Adjerid, M. Aiffa and J.E. Flaherty, High-order finite element methods for singularly-pertubed elliptic and parabolic problems (in preparation). · Zbl 0827.65097 [2] Adjerid, S.; Flaherty, J.E., Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems, Numer. math., 53, 183-198, (1988) · Zbl 0628.65104 [3] Adjerid, S.; Flaherty, J.E.; Moore, P.K.; Wang, Y.J., High-order adaptive methods for parabolic systems, Phys. D, 60, 94-111, (1992) · Zbl 0790.65088 [4] Adjerid, S.; Flaherty, J.E.; Wang, Y., A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems, () · Zbl 0791.65070 [5] Arney, D.C.; Flaherty, J.E., An adaptive local mesh refinement method for time-dependent partial differential equations, Appl. numer. math., 5, 257-274, (1989) · Zbl 0675.65119 [6] Babuška, I., The p- and hp-versions of the finite element method: the state of the art, () · Zbl 0665.73060 [7] Baehmann, P.L.; Wittchen, S.L.; Shephard, M.S.; Grice, K.R.; Yerry, M.A., Robust, geometrically based, automatic two-dimensional mesh generation, Internat. J. numer. methods engrg., 24, 1043-1078, (1987) · Zbl 0618.65116 [8] Biswas, R., Parallel and adaptive methods for hyperbolic partial differential systems, () [9] Brooks, A.N.; Hughes, T.J.R., Streamline upwind⧸petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041 [10] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. comp., 54, 545-581, (1990) · Zbl 0695.65066 [11] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. comput. phys., 84, 90-113, (1989) · Zbl 0677.65093 [12] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. comp., 52, 411-435, (1989) · Zbl 0662.65083 [13] Hughes, T.J.R.; Franca, L.P.; Mallet, M.; Mizukami, A.; Hughes, T.J.R.; Franca, L.P.; Mallet, M.; Mizukami, A.; Hughes, T.J.R.; Franca, L.P.; Mallet, M.; Mizukami, A.; Hughes, T.J.R.; Franca, L.P.; Mallet, M.; Mizukami, A., A new finite element formulation for computational fluid dynamics, IV, Comput. methods appl. mech. engrg., Comput. methods appl. engrg., Comput. methods appl. mech. engrg., Comput. methods appl. mech. engrg., 58, 329-336, (1986) · Zbl 0572.76068 [14] Lafon, F.; Osher, S., High-order filtering methods for approximating hyperbolic systems of conservation laws, ICASE report no. 90-25, (1990), Hampton, VA · Zbl 0746.65070 [15] Rank, E.; Babuška, I., An expert system for the optimal mesh design in the hp-version of the finite element method, Internat. J. numer. methods engrg., 24, 2087-2106, (1987) · Zbl 0621.73099 [16] Richtmyer, R.D.; Morton, K.W., Difference methods for initial value problems, (1967), Interscience New York · Zbl 0155.47502 [17] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066 [18] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. sci. statist. comput., 9, 1073-1084, (1988) · Zbl 0662.65081 [19] Shu, C.-W., TVB boundary treatment for numerical solutions of conservative laws, Math. comp., 49, 123-134, (1987) · Zbl 0628.65076 [20] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. comput. phys., 83, 32-78, (1989) · Zbl 0674.65061 [21] Sod, G.A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. comput. phys., 27, 1-31, (1978) · Zbl 0387.76063 [22] Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. numer. anal., 21, 995-1011, (1984) · Zbl 0565.65048 [23] Szabo, B.; Babuška, I., Introduction to finite element analysis, (1990), Wiley New York [24] van Leer, B., Towards the ultimate conservative difference scheme, IV: a new approach to numerical convection, J. comput. phys., 23, 276-299, (1977) · Zbl 0339.76056
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.