AbstractsMathematics

This thesis focuses on the constructions and applications of (piecewise) tensor product wavelet bases for solving PDEs by the adaptive wavelet Galerkin method (awgm). Locally supported biorthogonal wavelets are constructed on the unit interval w.r.t. which second-order constant coefficient differential operators are sparse. The awgm is applied for the numerical solution of singularly perturbed problems on the square. It will be shown that the awgm produces approximations that converge in energy norm with the best possible rate. A simultaneous space-time variational formulation of a parabolic evolution problem is solved with the awgm. This method is shown to converge with the best possible rate in linear complexity. Temporal test and trial wavelets are constructed such that the bi-infinite stiffness matrices of parabolic problems w.r.t. the tensor product wavelets are truly sparse. We construct a basis for a range of Sobolev spaces on a domain Ω from corresponding bases on subdomains that form a non-overlapping decomposition. We prove approximation rates from the resulting piecewise tensor product basis that are independent of the spatial dimension of Ω. The dimension independent rates will be realized numerically in linear complexity by the awgm. We study second-order linear elliptic problems with discontinuous diffusion coefficients. A domain decomposition technique is used to construct a piecewise tensor product wavelet basis that, when normalised w.r.t. the energy-norm, has Riesz constants that are bounded uniformly in the jumps. An awgm is applied to solve the boundary value problem with the optimal rate from the basis.