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Adaptive tensor product wavelet methods are applied for solving Poisson’s equation, as well as anisotropic generalizations, in high space dimensions. It will be demonstrated that the resulting approximations converge in energy norm with the same rate as the best approximations from the span of the best N tensor product wavelets, where moreover the constant factor that we may lose is independent of the space dimension n. The cost of producing these approximations will be proportional to their length with a constant factor that may grow with n, but only linearly.

We construct a wavelet basis on the unit interval with respect to which both the (infinite) mass and stiffness matrix corresponding to the one-dimensional Laplacian are (truly) sparse and boundedly invertible. As a consequence, the (infinite) stiffness matrix corresponding to the Laplacian on the n-dimensional unit box with respect to the n-fold tensor product wavelet basis is also sparse and boundedly invertible. This greatly simplifies the implementation and improves the quantitative properties of an adaptive wavelet scheme to solve the multi-dimensional Poisson equation. The results extend to any second order partial differential operator with constant coefficients that defines a boundedly invertible operator.