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, 35R05

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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)
 

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