Anisotropic wavelets along vector fields and applications to pde’s

Abstract EN

We start from tensor-product anisotropic wavelets on the n-dimensional unit cube.

Using an appropriate domain decomposition and mapping strategy, we build a system of wavelets in a bounded domain Ω of IRn.

They are anisotropic along the integral curves of a regular vector field defined in Ω, and they allow the representation of primal and dual norms of anisotropic Sobolev spaces.

We apply these results to define a non-standard mathematical formulation of convection–diffusion problems, which is the infinite-dimensional counterpart of the well-known SUPG stabilized formulation in finite elements.

We obtain a control over both the exact solution and its Galerkin approximation, in a norm which measures the smoothness of order 1/2 along the streamline directions and which is uniform in the diffusion parameter.

Convergence results in the same norm are also established.