Accurate gradient computations at interfaces using finite element methods

Abstract New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution, but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea of Wheeler (1974). For 2D interface problems, the point is to introduce a small tube near the interface and propose the gradient as part of unknowns, which is similar to a mixed finite element method, but only at the interface. Thus the computational cost is just slightly higher than in the standard finite element method. We present a rigorous one dimensional analysis, which shows a second order convergence order for both the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.

[1]  L. Wahlbin Superconvergence in Galerkin Finite Element Methods , 1995 .

[2]  L. Tartar An Introduction to Sobolev Spaces and Interpolation Spaces , 2007 .

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

[4]  Xu Yang,et al.  Gradient recovery for elliptic interface problem: II. Immersed finite element methods , 2016, J. Comput. Phys..

[5]  Do Y. Kwak,et al.  Optimal convergence analysis of an immersed interface finite element method , 2010, Adv. Comput. Math..

[6]  Tao Lin,et al.  Partially Penalized Immersed Finite Element Methods For Elliptic Interface Problems , 2015, SIAM J. Numer. Anal..

[7]  So-Hsiang Chou,et al.  An Immersed Linear Finite Element Method with Interface Flux Capturing Recovery , 2012 .

[8]  Christian Rey,et al.  The finite element method in solid mechanics , 2014 .

[9]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[10]  Jin-yi Sun,et al.  Positive solutions of second-order differential systems with nonhomogeneous boundary conditions , 2012 .

[11]  Joseph E. Pasciak The Mathematical Theory of Finite Element Methods (Susanne C. Brenner and L. Ridgway Scott) , 1995, SIAM Rev..

[12]  Xiaoming He,et al.  Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions , 2011 .

[13]  Error Estimates for the Finite Element Method , 2002 .

[14]  Zhilin Li The immersed interface method using a finite element formulation , 1998 .

[15]  Anna Karczewska,et al.  A finite element method for extended KdV equations , 2016, Int. J. Appl. Math. Comput. Sci..

[16]  Xingzhou Yang,et al.  The immersed interface method for elasticity problems with interfaces , 2002 .

[17]  Ivo Babuska,et al.  The finite element method for elliptic equations with discontinuous coefficients , 1970, Computing.

[18]  G. Burton Sobolev Spaces , 2013 .

[19]  Bo Li,et al.  Immersed-Interface Finite-Element Methods for Elliptic Interface Problems with Nonhomogeneous Jump Conditions , 2007, SIAM J. Numer. Anal..

[20]  Mary F. Wheeler,et al.  A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems , 1974 .

[21]  James H. Bramble,et al.  A finite element method for interface problems in domains with smooth boundaries and interfaces , 1996, Adv. Comput. Math..

[22]  J. Kevorkian,et al.  Partial Differential Equations: Analytical Solution Techniques , 1990 .

[23]  Kye T. Wee,et al.  An Analysis of a Broken P1-Nonconforming Finite Element Method for Interface Problems , 2009, SIAM J. Numer. Anal..

[24]  Carsten Carstensen,et al.  Low-order dPG-FEM for an elliptic PDE , 2014, Comput. Math. Appl..

[25]  Tao Lin,et al.  Linear and bilinear immersed finite elements for planar elasticity interface problems , 2012, J. Comput. Appl. Math..

[26]  Mary F. Wheeler,et al.  A Galerkin Procedure for Estimating the Flux for Two-Point Boundary Value Problems , 1974 .

[27]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[28]  Zhilin Li,et al.  A new augmented immersed finite element method without using SVD interpolations , 2015, Numerical Algorithms.

[29]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[30]  J. Zou,et al.  Finite element methods and their convergence for elliptic and parabolic interface problems , 1998 .

[31]  F. H. Soward Book Review: The Year Book of World Affairs, 1953 , 1953 .

[32]  F. Flores,et al.  Interfaces in crystalline materials , 1994, Thin Film Physics and Applications.

[33]  O. C. Zienkiewicz,et al.  The Finite Element Method for Solid and Structural Mechanics , 2013 .

[34]  Waixiang Cao,et al.  Superconvergence of immersed finite element methods for interface problems , 2015, Adv. Comput. Math..

[35]  Na An,et al.  A partially penalty immersed interface finite element method for anisotropic elliptic interface problems , 2014 .