We present a new adaptive numerical scheme for solving parabolic PDEs in
cartesian geometry. Applying a finite volume discretization with explicit
time integration, both of second order, we employ a fully adaptive
multiresolution scheme to represent the solution on locally refined nested
grids. The fluxes are evaluated on the adaptive grid. A dynamical adaption
strategy to advance the grid in time and to follow the time evolution of
the solution directly explaoits the multiresolution representation.
Applying this new method to several test probelms in one, two and three
space dimensions, like convection-diffucion, viscous Burgers and
reaction-diffusion equations, we show its second order accuracy and
demonstrate its computational efficiency.
This work is joint work with Olivier Roussel.
In this talk I present my recent results on the regularity conditions for a solution to the 3D Navier-Stokes equations with powers of the Laplacian, which incorporates the vorticity direction and its magnitude simultaneously. For the proof of the we exploit geometric properties of the vortex stretching term as well as the estimate using the Triebel-Lizorkin type of norms.