A nonlinear Galerkin method in the collocation case

Dr. Lucia Dettori

Division of Applied Mathematics, Brown University and Dept. of Mathematics & Statistics, University of Maryland Baltimore County

In this talk we present the implementation of the Nonlinear Galerkin method with collocation discretizations based on Fourier and Chebyshev aproximations. The nonlinear Galerkin method is a multiresolution method stemming from the theory of inertial manifolds. It is based on decompositions of the unknown into two arrays of unknowns, which represent respectively the large scale component and the small scale component of the solution. Such a decomposition is accomplished via the splitting of the fine grid into two coarse grids. This produces interesting connections between the physical space and the Fourier/Chebyshev space representation of the function. The method is applied to a nonlinear parabolic equation; its stability is proved and implementation issues are discussed, showing the advantages of the method.