Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T21:38:40.019Z Has data issue: false hasContentIssue false
  • English
  • Français

HP-finite element approximationson non-matching gridsfor partial differential equations with non-negativecharacteristic form

Published online by Cambridge University Press:  15 March 2003

Andrea Toselli
Andrea Toselli*
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. toselli@sam.math.ethz.ch.
Get access

Abstract

We propose and analyze a domain decomposition method on non-matching gridsfor partial differential equations with non-negativecharacteristic form. No weak or strong continuity of the finiteelement functions, their normal derivatives, or linearcombinations of the two is imposed across the boundaries of the subdomains.Instead, we employ suitable bilinear forms defined on the commoninterfaces, typical of discontinuous Galerkinapproximations.We prove an error bound which is optimal with respect to the mesh–size andsuboptimal with respect to the polynomial degree.Our analysis is valid for arbitrary shape–regular meshes and arbitrarypartitions into subdomains.Our method can be applied to advective, diffusive, and mixed–type equations,as well,and is well-suited for problems coupling hyperbolic and elliptic equations.We present some two-dimensionalnumerical results that support our analysis for the case of linear finite elements.

Keywords

Advection–diffusionhyperbolic problemsstabilizationdomain decompositionnon-matching gridsdiscontinuous Galerkinhp-finite elements.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 1 , January 2003 , pp. 91 - 115
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achdou, Y., The mortar element method for convection diffusion problems. C.R. Acad. Sci. Paris Sér. I Math. 321 (1995) 117-123.
Achdou, Y., Japhet, C., Maday, Y. and Nataf, F., A new cement to glue non-conforming grids with Robin interface conditions: The finite volume case. Numer. Math. 92 (2002) 593-620. CrossRef
Arbogast, T., Cowsar, L.C., Wheeler, M.F. and Yotov, I., Mixed finite element methods on non-matching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295-1315. CrossRef
Arbogast, T. and Yotov, I., A non-mortar mixed finite element method for elliptic problems on non-matching multiblock grids. Comput. Methods Appl. Mech. Engrg. 149 (1997) 255-265. CrossRef
Babuska, I. and Suri, M., The hp version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér. 21 (1987) 199-238. CrossRef
R. Becker and P. Hansbon, A finite element method for domain decomposition with non-matching grids. Technical Report N° 3613, INRIA, January 1999.
Ben Belgacem, F. and Maday, Y., The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289-302. CrossRef
Ben Belgacem, F. and Maday, Y., Coupling spectral and finite element for second order elliptic three dimensional equations. SIAM J. Numer. Anal. 31 (1999) 1234-1263. CrossRef
A. Ben Abdallah, F. Ben Belgacem, Y. Maday and F. Rapetti, Mortaring the two-dimensional Nédélec finite element for the discretization of the Maxwell equations. M 2 AS (submitted).
Ben Belgacem, F., The mortar element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197. CrossRef
C. Bernardi, Y. Maday and A.T. Patera, A new non conforming approach to domain decomposition: The mortar element method, in Collège de France Seminar, H. Brezis and J.-L. Lions Eds., Pitman (1994).
Bey, K.S., Patra, A. and Oden, J.T., hp-version discontinuous Galerkin methods for hyperbolic conservation laws: A parallel adaptive strategy. Internat. J. Numer. Methods Engrg. 38 (1995) 3889-3908. CrossRef
Brezzi, F. and Marini, D., A three-field domain decomposition method, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, A. Quarteroni, Y.A. Kuznetsov, J. Périaux and O.B. Widlund Eds., AMS. Contemp. Math. 157 (1994) 27-34. Held in Como, Italy, June 15-19, 1992. CrossRef
Brezzi, F. and Marini, D., Error estimates for the three-field formulation with bubble functions. Math. Comp. 70 (2001) 911-934. CrossRef
B. Cockburn, G.E. Karniadakis and Chi-Wang Shu (Eds.), Discontinuous Galerkin Methods. Springer-Verlag, Lect. Notes Comput. Sci. Eng. 11 (2000).
Houston, P., Schwab, C. and Süli, E., Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002) 2133-2163. CrossRef
Houston, P. and Süli, E., Stabilised hp-finite element approximation of partial differential equations with nonnegative characteristic form. Computing 66 (2001) 99-119. Archives for scientific computing. Numerical methods for transport-dominated and related problems, Magdeburg (1999). CrossRef
Hughes, T.J.R., Franca, L.P. and Hulbert, G.M., A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg. 73 (1989) 173-189. CrossRef
C. Johnson, Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987).
Johnson, C., Nävert, U. and Pitkäranta, J., Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1984) 285-312. CrossRef
Johnson, C. and Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1-26. CrossRef
Le Tallec, P. and Sassi, T., Domain decomposition with nonmatching grids: Augmented Lagrangian approach. Math. Comp. 64 (1995) 1367-1396. CrossRef
A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer-Verlag, Berlin (1994).
C. Schwab, p - and hp -finite element methods. Oxford Science Publications (1998).
R. Stenberg, Mortaring by a method of J.A. Nitsche, in Computational Mechanics: New trends and applications, S. Idelshon, E. Onate and E. Dvorkin Eds., Barcelona (1998). @CIMNE.
Wheeler, M.F. and Yotov, I., Physical and computational domain decompositions for modeling subsurface flows, in Tenth International Conference on Domain Decomposition Methods, J. Mandel, C. Farhat and X.-C. Cai Eds., AMS. Contemp. Math. 218 (1998) 217-228. CrossRef
Wohlmuth, B.I., A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38 (2000) 989-1012. CrossRef
I. Yotov, Mixed Finite Element Methods for Flow in Porous Media. Ph.D. thesis, TICAM, University of Texas at Austin (1996).
Yotov, I., A mixed finite element discretization on non-matching multiblock grids for a degenerate parabolic equation arising in porous media flow. East-West J. Numer. Math. 5 (1997) 211-230.