- Volume 24 Issue 2
DOI QR Code
A CONSISTENT DISCONTINUOUS BUBBLE SCHEME FOR ELLIPTIC PROBLEMS WITH INTERFACE JUMPS
- KWONG, IN (SAMSUNG ELECTRONICS SEMICONDUCTOR R & D CENTER) ;
- JO, WANGHYUN (DEPARTMENT OF MATHEMATICS, KUNSAN NATIONAL UNIVERSITY)
- Received : 2020.03.24
- Accepted : 2020.05.27
- Published : 2020.06.25
We propose a consistent numerical method for elliptic interface problems with nonhomogeneous jumps. We modify the discontinuous bubble immersed finite element method (DB-IFEM) introduced in (Chang et al. 2011), by adding a consistency term to the bilinear form. We prove optimal error estimates in L2 and energy like norm for this new scheme. One of the important technique in this proof is the Bramble-Hilbert type of interpolation error estimate for discontinuous functions. We believe this is a first time to deal with interpolation error estimate for discontinuous functions. Numerical examples with various interfaces are provided. We observe optimal convergence rates for all the examples, while the performance of early DB-IFEM deteriorates for some examples. Thus, the modification of the bilinear form is meaningful to enhance the performance.
- K. GARRETT AND H. ROSENBERG, The thermal conductivity of epoxy-resin/powder composite materials, Journal of Physics D: Applied Physics, 7 (1974), p. 1247.
- T. BELYTSCHKO, N. MOES, S. USUI, AND C. PARIMI, Arbitrary discontinuities in finite elements, International Journal for Numerical Methods in Engineering, 50 (2001), pp. 993-1013.
- Z. HASHIN, Thin interphase/imperfect interface in elasticity with application to coated fiber composites, Journal of the Mechanics and Physics of Solids, 50 (2002), pp. 2509-2537.
- A. BUFFA, Remarks on the discretization of some noncoercive operator with applications to heterogeneous maxwell equations, SIAM Journal on Numerical Analysis, 43 (2005), pp. 1-18.
- Z. CHEN, Reservoir simulation: mathematical techniques in oil recovery, SIAM, 2007.
- H. DUAN AND B. L. KARIHALOO, Effective thermal conductivities of heterogeneous media containing multiple imperfectly bonded inclusions, Physical Review B, 75 (2007), p. 064206.
- F. PAVANELLO, F. MANCA, P. LUCA PALLA, AND S. GIORDANO, Generalized interface models for transport phenomena: Unusual scale effects in composite nanomaterials, Journal of Applied Physics, 112 (2012), p. 084306.
- R. PLONSEY, Bioelectric sources arising in excitable fibers (Alza lecture), Annals of biomedical engineering, 16 (1988), pp. 519-546.
- M. R. HOSSAN, R. DILLON, AND P. DUTTA, Hybrid immersed interface-immersed boundary methods for ac dielectrophoresis, Journal of Computational Physics, 270 (2014), pp. 640-659.
- G. CHAVENT AND J. JAFFRE, Mathematical models and finite elements for reservoir simulation: single phase, multiphase and multicomponent flows through porous media, Elsevier, 1986.
- C. VAN DUIJN, J. MOLENAAR, AND M. DE NEEF, The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media, Transport in Porous Media, 21 (1995), pp. 71-93.
- M. F. WHEELER, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), pp. 152-161.
- B. COCKBURN, G. E. KARNIADAKIS, AND C.-W. SHU, The development of discontinuous Galerkin methods, in Discontinuous Galerkin Methods, Springer, 2000, pp. 3-50.
- D. N. ARNOLD, F. BREZZI, B. COCKBURN, AND L. D. MARINI, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM journal on numerical analysis, 39 (2002), pp. 1749-1779.
- A. ERN, I. MOZOLEVSKI, AND L. SCHUH, Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures, Computer methods in applied mechanics and engineering, 199 (2010), pp. 1491-1501.
- N. MOE S, J. DOLBOW, AND T. BELYTSCHKO, A finite element method for crack growth without remeshing, International journal for numerical methods in engineering, 46 (1999), pp. 131-150.
- T. BELYTSCHKO AND T. BLACK, Elastic crack growth in finite elements with minimal remeshing, International journal for numerical methods in engineering, 45 (1999), pp. 601-620.
- P. KRYSL AND T. BELYTSCHKO, An efficient linear-precision partition of unity basis for unstructured meshless methods, Communications in Numerical Methods in Engineering, 16 (2000), pp. 239-255.
- T. BELYTSCHKO, C. PARIMI, N. MOES, N. SUKUMAR, AND S. USUI, Structured extended finite element methods for solids defined by implicit surfaces, International journal for numerical methods in engineering, 56 (2003), pp. 609-635.
- G. LEGRAIN, N. MOES, AND E. VERRON, Stress analysis around crack tips in finite strain problems using the extended finite element method, International Journal for Numerical Methods in Engineering, 63 (2005), pp. 290-314.
- Z. LI, T. LIN, AND X. WU, New cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik, 96 (2003), pp. 61-98.
- Z. LI, T. LIN, Y. LIN, AND R. C. ROGERS, An immersed finite element space and its approximation capability, Numerical Methods for Partial Differential Equations, 20 (2004), pp. 338-367.
- S. H. CHOU, D. Y. KWAK, AND K. T. WEE, Optimal convergence analysis of an immersed interface finite element method, Advances in Computational Mathematics, 33 (2010), pp. 149-168.
D. Y. KWAK, K. T. WEE, AND K. S. CHANG, An analysis of a broken
$P_1$-nonconforming finite element method for interface problems, SIAM Journal on Numerical Analysis, 48 (2010), pp. 2117-2134.
D. Y. KWAK AND J. LEE, A modified
$P_1$-immersed finite element method, International Journal of Pure and Applied Mathematics, 104 (2015), pp. 471-494.
D. Y. KWAK, S. JIN, AND D. KYEONG, A stabilized
$P_1$-nonconforming immersed finite element method for the interface elasticity problems, ESAIM: Mathematical Modelling and Numerical Analysis, 51 (2017), pp. 187-207.
- D. KYEONG AND D. Y. KWAK, An immersed finite element method for the elasticity problems with displacement jump, Advances in Applied Mathematics and Mechanics, 9 (2017), pp. 407-428.
- S. JIN, D. Y. KWAK, AND D. KYEONG, A consistent immersed finite element method for the interface elasticity problems, Advances in Mathematical Physics, 2016 (2016).
- G. JO AND D. Y. KWAK, An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid, Computer Methods in Applied Mechanics and Engineering, (2017).
- D. Y. KWAK, S. LEE, AND H. YUNKYONG, A new finite element for interface problems having robin type jump, Inernational Journal of Numerical Analysis and Modeling, 14 (2017), pp. 532-549.
- M. CROUZEIX AND P. A. RAVIART, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, Revue francaise d'automatique, informatique, recherche operationnelle. Mathematique, 7 (1973), pp. 33-75.
- S. H. CHOU, D. Y. KWAK, AND K. Y. KIM, Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems, Mathematics of computation, 72 (2003), pp. 525-539.
- K. S. CHANG AND D. Y. KWAK, Discontinuous bubble scheme for elliptic problems with jumps in the solution, Computer Methods in Applied Mechanics and Engineering, 200 (2011), pp. 494-508.
- T. LIN, Q. YANG, AND X. ZHANG, A priori error estimates for some discontinuous Galerkin immersed finite element methods, Journal of Scientific Computing, 65 (2015), pp. 875-894.
- T. LIN, Y. LIN, AND X. ZHANG, Partially penalized immersed finite element methods for elliptic interface problems, SIAM Journal on Numerical Analysis, 53 (2015), pp. 1121-1144.
- J. A. ROITBERG ET AL., A theorem on homeomorphisms for elliptic systems and its applications, Mathematics of the USSR-Sbornik, 7 (1969), p. 439.
- J. H. BRAMBLE AND J. T. KING, A finite element method for interface problems in domains with smooth boundaries and interfaces, Advances in Computational Mathematics, 6 (1996), pp. 109-138.