$HP$-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form

Toselli, Andrea

ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 1, pp. 91-115.

We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect to the mesh-size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape-regular meshes and arbitrary partitions 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-dimensional numerical results that support our analysis for the case of linear finite elements.

[1] Y. Achdou, The mortar element method for convection diffusion problems. C.R. Acad. Sci. Paris Sér. I Math. 321 (1995) 117-123. | Zbl

[2] Y. Achdou, C. Japhet, Y. Maday and F. Nataf, A new cement to glue non-conforming grids with Robin interface conditions: The finite volume case. Numer. Math. 92 (2002) 593-620. | Zbl

[3] T. Arbogast, L.C. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on non-matching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295-1315. | Zbl

[4] T. Arbogast and I. Yotov, A non-mortar mixed finite element method for elliptic problems on non-matching multiblock grids. Comput. Methods Appl. Mech. Engrg. 149 (1997) 255-265. | Zbl

[5] I. Babuška and M. Suri, The $hp$ version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér. 21 (1987) 199-238. | Numdam | Zbl

[6] R. Becker and P. Hansbon, A finite element method for domain decomposition with non-matching grids

[7] F. Ben Belgacem and Y. Maday, The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289-302. | Numdam | Zbl

[8] F. Ben Belgacem and Y. Maday, Coupling spectral and finite element for second order elliptic three dimensional equations. SIAM J. Numer. Anal. 31 (1999) 1234-1263. | Zbl

[9] 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

[10] F. Ben Belgacem, The mortar element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197. | Zbl

[11] 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). | Zbl

[12] K.S. Bey, A. Patra and J.T. Oden, $hp$-version discontinuous Galerkin methods for hyperbolic conservation laws: A parallel adaptive strategy. Internat. J. Numer. Methods Engrg. 38 (1995) 3889-3908. | Zbl

[13] F. Brezzi and D. Marini, 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. | Zbl

[14] F. Brezzi and D. Marini, Error estimates for the three-field formulation with bubble functions. Math. Comp. 70 (2001) 911-934. | Zbl

[15] B. Cockburn, G.E. Karniadakis and Chi-Wang Shu (Eds.), Discontinuous Galerkin Methods. Springer-Verlag, Lect. Notes Comput. Sci. Eng. 11 (2000). | MR

[16] P. Houston, C. Schwab and E. Süli, Discontinuous $hp$-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002) 2133-2163. | Zbl

[17] P. Houston and E. Süli, 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). | Zbl

[18] T.J.R. Hughes, L.P. Franca and G.M. Hulbert, 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. | Zbl

[19] C. Johnson, Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987). | Zbl

[20] C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1984) 285-312. | Zbl

[21] C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1-26. | Zbl

[22] P. Le Tallec and T. Sassi, Domain decomposition with nonmatching grids: Augmented Lagrangian approach. Math. Comp. 64 (1995) 1367-1396. | Zbl

[23] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer-Verlag, Berlin (1994). | MR | Zbl

[24] C. Schwab, $p$- and $hp$-finite element methods. Oxford Science Publications (1998). | MR | Zbl

[25] 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. | MR

[26] M.F. Wheeler and I. Yotov, 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. | Zbl

[27] B.I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38 (2000) 989-1012. | Zbl

[28] I. Yotov, Mixed Finite Element Methods for Flow in Porous Media. Ph.D. thesis, TICAM, University of Texas at Austin (1996).

[29] I. Yotov, 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. | Zbl