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IMA Journal of Applied Mathematics Advance Access originally published online on December 15, 2008
IMA Journal of Applied Mathematics 2009 74(1):74-84; doi:10.1093/imamat/hxn040
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The residual velocity method applied to a steady free boundary-value problem of vector Laplacian type

Wan Chen{dagger} and Brian Wetton{ddagger}

Department of Mathematics, University of British Columbia #121 1984 Mathematics Rd, Vancouver, B.C., Canada V6T1Z2

{dagger} Email: wanchen{at}math.ubc.ca

{ddagger} Corresponding author. Email: wetton{at}math.ubc.ca.

Received on November 8, 2007; Revision received July 18, 2008. Accepted on November 7, 2008

We consider a free boundary-value problem based on a simplified model of two-phase flow in porous media. The model has two independent variables on each side of the free interface. At the interface at steady state, five mixed Dirichlet and Neumann conditions are given. The movement of the interface in time-dependent situations can be reduced to a normal motion proportional to the residual in one of the steady-state interface conditions (the elliptic interior problems and the other interface conditions are satisfied at each time). Following previous work, we consider the use of other residuals for the normal velocity that have superior numerical properties. The well-posedness criteria for this vector example are particularly clear. The advantages of the correctly chosen, non-physical residual velocities are demonstrated in numerical computations. Although the finite-difference implementation in this work is not applicable to general problems, it has superior performance to previous implementations.

Keywords: well-posedness; free boundary problem; residual velocity.


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