Adaptive Wavelet Methods for Elliptic Operator Equations with Nonlinear Terms

We present an adaptive wavelet method for the numerical solution of elliptic operator equations with nonlinear terms. This method is developed based on tree approximations for the solution of the equations and adaptive fast reconstruction of nonlinear functionals of wavelet expansions. We introduce a constructive greedy scheme for the construction of such tree approximations. Adaptive strategies of both continuous and discrete versions are proposed. We prove that these adaptive methods generate approximate solutions with optimal order in both of convergence and computational complexity when the solutions have certain degree of Besov regularity.