Commun. Comput. Phys., 6 (2009), pp. 185-202. |
A Bilinear Immersed Finite Volume Element Method for the Diffusion Equation with Discontinuous Coefficient X.-M. He ^{1}, T. Lin ^{1*}, Y. Lin ^{2} 1 Department of Mathematics, Virginia Tech, Blacksburg, VA 24060, USA.2 Department of Mathematical and Statistical Science, University of Alberta, Edmonton AB, T6G 2G1, Canada; and Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong. Received 17 December 2007; Accepted (in revised version) 6 June 2008 Available online 24 November 2008 Abstract This paper is to present a finite volume element (FVE) method based on the bilinear immersed finite element (IFE) for solving the boundary value problems of the diffusion equation with a discontinuous coefficient (interface problem). This method possesses the usual FVE method's local conservation property and can use a structured mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient has discontinuity along piecewise smooth nontrivial curves. Numerical examples are provided to demonstrate features of this method. In particular, this method can produce a numerical solution to an interface problem with the usual O(h^2) (in L^2 norm) and O(h) (in H^1 norm) convergence rates. AMS subject classifications: 65N15, 65N30, 65N50, 35R05Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Interface problems, immersed interface, finite volume element, discontinuous coefficient, diffusion equation. *Corresponding author. Email: xiaoming@vt.edu (X.-M. He), tlin@vt.edu (T. Lin), yanlin@ualberta.ca (Y. Lin) |