Space-time sparse discretization of linear parabolic equations

Roman Andreev, ETH Z├╝rich

Abstract: For a class of linear parabolic equations we propose a nonadaptive sparse wavelet finite element space-time discretization motivated by the formulation presented in [Ch. Schwab, R. Stevenson, "Space-time adaptive wavelet methods for parabolic evolution problems", MCOM, 78, 2009]. The problem is reduced to a finite, overdetermined linear system of equations. We prove stability, i.e., that the finite section normal equations are well-conditioned if appropriate Riesz bases are employed, and that the Galerkin solution converges quasi-optimally in the natural solution space for the original equation. Numerical examples confirm the theory. Extension to parabolic problems with stochastic coefficients is considered.