Defect corrected averaging for parabolic PDEs

Extended defect corrected averaging for highly oscillatory PDEs

We consider parabolic partial differential equations where the support of the sources or of the inhomogenous Neumann boundary conditions undergoes an oscillatory motion. The timescale of this oscillation is much smaller than the characteristic time scale of diffusion, which is of main interest. Resolution of the smallest timescale constitutes a strong restriction on the stepsize of time integra...

Multirate Numerical Integration for Parabolic PDEs

To solve PDE problems with different time scales that are localized in space, multirate time integration is examined. This technique enables one to use large time steps for slowly time-varying spatial regions, and small steps for rapidly varying ones. Multirate time stepping is coupled with the local uniform grid refinement and provides a robust and efficient method for the target problem class...

Infinite dimensional backstepping for nonlinear parabolic PDEs

Completion of overdetermined parabolic PDEs

In this paper we apply methods of commutative algebra to analysis of systems of PDEs. More precisely, we show that systems which are parabolic in a generalized sense are equivalent to certain completed systems which are parabolic in the standard sense. We also propose a constructive method to get this completion, and Gröbner basis methods, via symbol modules of the systems, play a central role ...

Averaging for Fundamental Solutions of Parabolic Equations

Herein, an averaging theory for the solutions to Cauchy initial value problems of arbitrary order, "-dependent parabolic partial di erential equations is developed. Indeed, by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the fundamental solution of an \averaged" parabolic equation, we bring forth a novel approach to compar...