Convergence of Lowest Order Semi-Lagrangian Schemes

• Published in: Foundations of computational mathematics Vol. 13; no. 2; pp. 187 - 220
• Main Authors: Heumann, Holger, Hiptmair, Ralf
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.04.2013; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Applications of Mathematics; Approximation; Asymptotic properties; Boundary conditions; Computational mathematics; Computer Science; Convergence; Convergence (Mathematics); Differential equations; Discretization; Economics; Estimates; Foundations; Galerkin methods; Lagrangian functions; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Switzerland; Advection-diffusion problem; 65m60; 65m25; Semi-Lagrangian methods; 65m12; Discrete differential forms
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-012-9139-3

We consider generalized linear transient advection-diffusion problems for differential forms on a bounded domain in ℝ d . We provide comprehensive a priori convergence estimates for their spatiotemporal discretization by means of a first-order in time semi-Lagrangian approach combined with a discontinuous Galerkin method. Under rather weak assumptions on the velocity underlying the advection we establish an asymptotic L 2 -estimate of order , where h is the spatial meshwidth, τ denotes the time step, and r is the polynomial degree of the forms used as trial functions. This estimate can be improved considerably in a variety of special settings.