Abstract
In this paper we consider the numerical approximation of a class of second order elliptic boundary value problems with discontinuous and highly periodically oscillating coefficients. We apply both classical and modified finite volume methods for the approximate solution of this problem. Error estimates depending on $\varepsilon$ the parameter involved in the periodic homogenization are established. Numerical simulations for one-dimensional problem confirm the theorical results and also show that the modified scheme has a smaller constant of convergence than the classical scheme based on harmonic averaging for this class of equations.
Highlights
There are many practical computational problems with highly oscillatory solutions e.g. computation of flow in heterogeneous porous media for petroleum and groundwater reservoir simulation (see, e.g., (Hornung, 1997) and the bibliographies therein)
Error estimates depending on ε the parameter involved in the periodic homogenization are established
Numerical simulations for one-dimensional problem confirm the theorical results and show that the modified scheme has a smaller constant of convergence than the classical scheme based on harmonic averaging for this class of equations
Summary
There are many practical computational problems with highly oscillatory solutions e.g. computation of flow in heterogeneous porous media for petroleum and groundwater reservoir simulation (see, e.g., (Hornung, 1997) and the bibliographies therein). If a porous medium with a periodic structure is considered, with the size of the period is small enough compared to the size of the reservoir, and denoting their ratio by ε (0 < ε
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