On Strongly Coupled Linear Elliptic Systems with Application to Otolith Membrane Distortion

Abstract

Problem Statement: In this research, the author discussed the problems associated with the approximation of the mixed derivative terms appearing in strongly coupled linear elliptic systems by the finite difference method over irregular domains. To avoid the appearance of mixed derivative terms the author introduced a reformulation for the system through introducing a new dependent variable which adds one supplementary (simple) differential equation to the system but does not change its elliptic character. Approach: The basic idea in the reformulation is the direct generation of the Laplacian operator which has an efficient finite difference treatment. Results: Two finite difference formulae with symmetric appearance approximating the first order derivatives on curved boundaries up to O(h2) are established, that can be considered as a generalization to the well known central formula. Applications to the otolith membrane model have proved the reliability and efficiency of the present treatment in comparison with other methods. Conclusions/Recommendations: Although, this treatment has increased the number of algebraic equations approximating the system linearly 3n instead of 2n, the overall accuracy is increased quadratically. The band width of matrix of coefficients of the algebraic system is decreased and there is no need to interpolate along the diagonals due to the absence of mixed derivatives. The treatment is promising and other extensions are mentioned.