A Note on the Convergence Analysis of a Diffuse-domain Approach

Abstract We analyse the error behaviour of a diffuse-domain approximation of an elliptic differential equation. In one dimension and for a half-plane problem in two dimensions an approximation quality of order one in the interface parameter is shown. Some supporting numerical experiments are also presented.

[1]  Zhilin Li,et al.  Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions , 2009 .

[2]  W. Hackbusch,et al.  Composite finite elements for the approximation of PDEs on domains with complicated micro-structures , 1997 .

[3]  Xiangrong Li,et al.  SOLVING PDES IN COMPLEX GEOMETRIES: A DIFFUSE DOMAIN APPROACH. , 2009, Communications in mathematical sciences.

[4]  Axel Voigt,et al.  AMDiS: adaptive multidimensional simulations , 2007 .

[5]  M. Gelinsky,et al.  Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds--a modeling approach. , 2011, Journal of biomechanics.

[6]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[7]  Wouter-Jan Rappel,et al.  Computational model for cell morphodynamics. , 2010, Physical review letters.

[8]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[9]  Tao Lin,et al.  New Cartesian grid methods for interface problems using the finite element formulation , 2003, Numerische Mathematik.

[10]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[11]  O. C. Zienkiewicz,et al.  A new cloud-based hp finite element method , 1998 .

[12]  M. Stynes,et al.  Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems , 1996 .

[13]  F. Fenton,et al.  Modeling wave propagation in realistic heart geometries using the phase-field method. , 2005, Chaos.

[14]  J. Lowengrub,et al.  Two-phase flow in complex geometries: A diffuse domain approach. , 2010, Computer modeling in engineering & sciences : CMES.

[15]  Michael Holst,et al.  Green's Functions and Boundary Value Problems: Stakgold/Green's Functions , 2011 .

[16]  Stefan A. Sauter,et al.  Two-scale composite finite element method for Dirichlet problems on complicated domains , 2006, Numerische Mathematik.

[17]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[18]  E. Kamke.  Differentialgleichungen : Lösungsmethoden und Lösungen , 1977 .