A multiwavelet based on piecewise \(C^1\) fractal functions and related applications to differential equations (Q1275307)

Piecewise \(C^1\) multiwavelets are constructed using fractal interpolation techniques, then it is shown how to use such wavelets to solve elliptic boundary value problems. The multiwavelets that are constructed bear many advantages in relation to Daubechies wavelets, for example. Specifically, besides being piecewise \(C^1\), the short supports and orthogonality of these wavelets to quadratic functions translate into fast numerical algorithms with good approximation properties, while their symmetries or antisymmetries at boundary points translate into simple expansions at boundaries. Furthermore, the fractal interpolation construction means that inner products between the functions and their derivatives are easily computed and the interpolatory property makes the wavelets suitable for collocation methods. The author reviews the fractal interpolation techniques and multiresolution techniques needed to construct the wavelets, constructs them, then outlines how they are implemented to solve elliptic equations with vanishing boundary values numerically.