Solution of Multiscale Partial Differential Equations Using Wavelets

Wavelets are a powerful new mathematical tool which offers the possibility totreat in a natural way quantities characterized by several length scales. Inthis article we will show how wavelets can be used to solve partialdifferential equations which exhibit widely varying length scales and which aretherefore hardly accessible by other numerical methods. As a benchmarkcalculation we solve Poisson's equation for a 3-dimensional Uranium dimer. Thelength scales of the charge distribution vary by 4 orders of magnitude in thiscase. Using lifted interpolating wavelets the number of iterations isindependent of the maximal resolution and the computational effort thereforescales strictly linearly with respect to the size of the system