×

Numerical approximation of elliptic interface problems via isoparametric finite element methods. (English) Zbl 1369.65150

Summary: This paper is concerned with the numerical approximation of elliptic interface problems via isoparametric finite element methods. First, a convergence analysis is carried which asserts that optimal rates of convergence are recovered in both the energy and the \(L^2\)-norms. Subsequently, the efficiency of isoparametric finite elements for the problem at hand is also assessed via numerical experiments. Results both with smooth and piecewise regular interfaces are presented and discussed. The numerical predictions corroborate the theoretical results and they also indicate that second-order convergence is achieved in the \(L^\infty\)-norm. Comparisons between isoparametric finite elements, Lagrangian finite elements and the Immersed Interface Method are also performed.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Software:

Gmsh
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Le Bars, M.; Grae Worster, M., Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification, J. Fluid Mech., 550, 149-173 (2006) · Zbl 1097.76066
[2] Varsakelis, C.; Papalexandris, M. V., Low-Mach-number asymptotics for two-phase flows of granular materials, J. Fluid Mech., 669, 472-497 (2011) · Zbl 1225.76297
[3] Leveque, R. J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 1019-1044 (1994) · Zbl 0811.65083
[4] Huang, H.; Li, Z., Convergence analysis of the immersed interface method, IMA J. Numer. Anal., 19, 583-608 (1999) · Zbl 0940.65114
[5] Beale, J. T.; Layton, A. T., On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. Appl. Math. Comput. Sci., 1, 91-119 (2006) · Zbl 1153.35319
[6] Li, Z.; Ito, K., Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23, 339-361 (2001) · Zbl 1001.65115
[7] Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35, 230-254 (1998) · Zbl 0915.65121
[8] Berthelsen, P. A., A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, J. Comput. Phys., 197, 364-386 (2004) · Zbl 1052.65100
[9] Liu, X.-D.; Fedkiw, R. P.; Kang, M., A boundary condition capturing method for Poisson equation on irregular domains, J. Comput. Phys., 160, 151-178 (2000) · Zbl 0958.65105
[10] Fedkiw, R. P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152, 457-492 (1999) · Zbl 0957.76052
[11] Tornberg, A.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. Comput. Phys., 200, 462-488 (2004) · Zbl 1115.76392
[12] Zhou, Y. C.; Zhao, S.; Feig, M.; Wei, G. W., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys., 213, 1-30 (2006) · Zbl 1089.65117
[13] Yu, S. N.; Wei, G. W., Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities, J. Comput. Phys., 227, 602-632 (2007) · Zbl 1128.65103
[14] Yu, S.; Zhou, Y.; Wei, G. W., Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces, J. Comput. Phys., 224, 729-756 (2007) · Zbl 1120.65333
[15] Mu, L.; Wang, J.; Wei, G.; Ye, X.; Zhao, S., Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250, 106-125 (2013) · Zbl 1349.65472
[16] Babûska, I., The finite element method for elliptic equations with discontinuous coefficients, Computing, 5, 207-213 (1970) · Zbl 0199.50603
[17] Barrett, J. W.; Elliott, C. M., Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA J. Numer. Anal., 7, 283-300 (1987) · Zbl 0629.65118
[18] Chen, Z.; Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79, 175-202 (1998) · Zbl 0909.65085
[19] Li, J.; Melenk, J. M.; Wohlmuth, B.; Zou, J., Optimal a priori estimates for higher order finite elements for elliptic interface problems, Appl. Numer. Math., 60, 19-37 (2010) · Zbl 1208.65168
[20] Chan, K. H.; Zhang, K.; Zou, J., Spherical interface dynamos: mathematical theory, finite element approximation, and application, SIAM J. Numer. Anal., 44, 1877-1902 (2006) · Zbl 1323.76119
[21] Hiptmair, R.; Li, J.; Zou, J., Convergence analysis of finite element methods for H(curl; Omega)-elliptic interface problems, Numer. Math., 122, 557-578 (2012) · Zbl 1268.78020
[22] Szabo, B.; Babuska, I., Finite Element Analysis (1991), Wiley · Zbl 0792.73003
[23] Lenoir, M., Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal., 23, 562-580 (1986) · Zbl 0605.65071
[24] Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods (1994), Springer: Springer New York · Zbl 0804.65101
[25] Kellogg, R. B., On the Poisson equation with intersecting interfaces, Appl. Anal., 4, 101-129 (1975) · Zbl 0307.35038
[26] Bernardi, C., Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal., 26, 1212-1240 (1989) · Zbl 0678.65003
[27] Ciarlet, P. G.; Raviart, P. A., Interpolation theory over curved elements, with applications to finite element methods, Comput. Methods Appl. Mech. Engrg., 1, 217-249 (1972) · Zbl 0261.65079
[28] Geuzaine, C.; Remacle, J.-F., Gmsh: a three-dimensional finite element mesh generator with build-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79, 1309-1331 (2009) · Zbl 1176.74181
[29] Ladyzhenskaya, O. A.; Rivkind, V. J.; Uralceva, N. N., The classical solvability of diffraction problems, Tr. Mat. Inst. Steklova, 92, 116-146 (1966) · Zbl 0165.11802
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.