A SPARSE LAPLACIAN IN TENSOR PRODUCT WAVELET COORDINATES

Abstract. 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 (infi-nite) 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 quantita-tive 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. 1