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Convergence of lowest-order semi-Lagrangian schemes. (English) Zbl 1273.65121
The authors consider a non-stationary advection-diffusion problem for time-dependent differential forms. By means of the Hille-Yosida theorem, the existence and the uniqueness of the transient advection-diffusion problem are obtained. The semi-Lagrangian Galerkin time-stepping scheme for the considered advection-diffusion problem is presented. Under some additional assumptions, an \(L^2\)-estimate of order \(O(\tau+h^r+h^{r+1}\tau^{-1/2}+\tau^{1/2})\) is established, with \(h\) the spatial meshsize, \(\tau\) the time step and \(r\) the polynomial degree of the trial functions.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
62M20 Inference from stochastic processes and prediction
35K20 Initial-boundary value problems for second-order parabolic equations
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