Explicit formulas for wavelet-homogenized coefficients of elliptic operators

Wavelet-based homogenization provides a method for constructing a coarse-grid discretization of a variable–coefficient differential operator that implicitly accounts for the influence of the fine scale medium parameters on the coarse scale of the solution. The method is applied to discretizations of operators of the form in one dimension and μ(x)Δ in one and more dimensions. The resulting homogenized matrices are shown to correspond to differential operators of the same (or closely related) form. In dimension one, results are obtained for periodic two-phase and for arbitrary coefficients μ(x). For periodic two-phase coefficients, the homogenized coefficients may be computed exactly as the harmonic mean of the function μ. For non-periodic coefficients, the “mass-lumping” approximation results in an explicit formula for the homogenized coefficients. In higher dimensions, results are obtained for operators of the form μ(x)Δ, where μ(x) may or may not be periodic; explicit formulae for the homogenized coefficients are also derived. Numerical examples in 1D and 2D are also presented.