On the computation of the stabilized coefficients for the 1D spectral VMS method

In this work, we study the computation of the stabilized coefficients for the Variational Multi-Scale method with spectral approximation of the sub-scales, applied to 1D problems. The method is based on an extension of the spectral theorem to operators that have an associated base of eigenfunctions, which are orthonormal in weighted L 2 spaces. We study the discretization of both second order elliptic and parabolic problems with the finite element method. The spectral VMS method is characterized as a standard VMS method with stabilized coefficients issued form the eigenfunctions of the sub-grid problem, that are computed analytically. We derive an off-line/on-line strategy for the computation of the stabilized coefficients. This allows a fast solution of the spectral VMS method, similar to that of the standard VMS one. We display some numerical tests for the stationary and evolutive one-dimensional advection–diffusion equations, in which observe super-convergence effects at the grid nodes.