| Adaptive wavelet methods for solving PDE's converge with the best possible rate in the basis that is used, in linear complexity. Moreover, these methods offer some important advantages that seem not be realizable with other approaches. When the equation is posed on $(0,1)^n$, tensor product bases can be applied. With such bases, the convergence rates are {\em independent of $n$}. Secondly, in three and more dimensions, solutions of PDE's are known to have limited smoothness in the scale of Besov spaces that govern the rate of convergence with standard bases. Best $N$-term approximation from tensor product bases provides {\em anisotropic refinements} towards the boundary, and therefore yields the optimal, moreover dimension independent rates. Finally, adaptive wavelet schemes can also be applied to {\em parabolic equations} giving optimal adaptivity {\em simultaneously} in space and time. Moreover, since the basis will be a tensor product of bases of temporal and spatial wavelets, the computational cost will be of the order of solving one stationary problem. The first goal in this project is to extend the advantage of adaptive tensor product approximations to non-product domains by using {\em piecewise tensor product approximation} via domain decomposition techniques. The second goal will be the extension of the resulting adaptive solver to {\em nonlinear variational problems}. The third goal will be to create an efficient implementation of the adaptive solver and to perform numerical tests to validate its practical performance. |
