Using an overlapping domain decomposition technique, suitable frames can easily be constructed and implemented. In a first step, we show that classical results on the convergence rates of best N-term approximations of the solution with respect to wavelet Riesz bases essentially carry over to the considered class of wavelet frames. We then develop an adaptive method based on a steepest descent iteration for the frame coordinate representation of the elliptic equation, and, most importantly, we develop algorithms based on multiplicative and additive Schwarz overlapping domain decomposition methods. We prove that our adaptive schemes are of asymptotically optimal complexity, in the sense that they realize the same convergence rate as the sequence of best N-term frame approximations of the solution. Moreover, using special numerical quadrature rules for the computation of the frame representation of the elliptic operator, the overall computational cost stays proportional to the number of wavelets selected by the algorithms.
The results of a series of numerical tests for non-trivial one- and two-dimensional Poisson and biharmonic model problems confirm our theoretical findings and particularly demonstrate the efficiency of the domain decomposition approach. A comparison with a standard adaptive finite element solver shows that our multiplicative Schwarz method potentially generates significantly sparser approximations. In addition, a parallel implementation of the new adaptive additive Schwarz wavelet solver is developed and tested.