Abstract
An a posteriori error estimate is derived for the approximation of the transport equation with a time dependent transport velocity. Continuous, piecewise linear, anisotropic finite elements are used for space discretization, the Crank-Nicolson scheme scheme is proposed for time discretization. This paper is a generalization of Dubuis S, Picasso M (J Sci Comput 75(1):350–375, 2018) where the transport velocity was not depending on time. The a posteriori error estimate (upper bound) is shown to be sharp for anisotropic meshes, the involved constant being independent of the mesh aspect ratio. A quadratic reconstruction of the numerical solution is introduced in order to obtain an estimate that is order two in time. Error indicators corresponding to space and time are proposed, their accuracy is checked with non-adapted meshes and constant time steps. Then, an adaptive algorithm is introduced, allowing to adapt the meshes and time steps. Numerical experiments are presented when the exact solution has strong variations in space and time, illustrating the efficiency of the method. They indicate that the effectivity index is close to one and does not depend on the solution, mesh size, aspect ratio, and time step.
Highlights
Space-time adaptive algorithms are efficient tools to approximate solutions of partial differential equations with accuracy and low computational cost
The classical theory of a posteriori error analysis for finite element methods was first developed on isotropic meshes [6, 16, 40], the involved constants were depending on the mesh aspect ratio
To recover an a posteriori error estimate for the Crank-Nicolson method applied to the heat equation which was of order two in time
Summary
Space-time adaptive algorithms are efficient tools to approximate solutions of partial differential equations with accuracy and low computational cost. The isotropic theory for a posteriori error estimates was updated, see for instance [4, 19, 22, 23, 25], and the involved constants were proved to be aspect ratio independent whenever the mesh was aligned with the solution. To circumvent this problem, piecewise quadratic reconstructions have been introduced for parabolic problems [1, 2, 7, 28], in [21] for the transport equations and in [27] for the wave equation. The a posteriori error analysis of [21] for the transport equation is extended to the case where the velocity varies in space and time.
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