An adaptive algorithm for the transport equation with time dependent velocity
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An adaptive algorithm for the transport equation with time dependent velocity Samuel Dubuis1 · Marco Picasso1 Received: 9 January 2020 / Accepted: 29 July 2020 © The Author(s) 2020 OPEN
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. Keywords A posteriori error estimates · Space-time adaptive algorithm · Anisotropic finite elements · Second order time discretization · Transport equation Mathematics Subject Classification 65M12 · 65M50
1 Introduction Space-time adaptive algorithms are efficient tools to approximate solutions of partial differential equations with accuracy and low computational cost. Whenever possible, the adaptive criteria is based on theoretical error estimates, this is mostly the case for elliptic and parabolic problems, fewer results are available for hyperbolic problems[11, 14, 18, 31, 37], nonlinear systems [3, 13, 30, 39, 41] or PDEs with variable coefficients [9, 24]. 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. However, anisotropic finite elements, that is to say elements with possibly large aspect ratio, have been widely used to approximate phenomena involving boundary or internal layers. The isotropic theory for a posteriori error estimates was therefore 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. The Crank-Nicolson method is a popular second order scheme for time dependent problems. However, most of the a posteriori error estimates are proved for first order methods only, for instance the Backward Euler scheme [8, 10, 32, 38]. Moreover, standard a posteriori proofs
* Marco Picasso, [email protected]; Samuel Dubuis, [email protected] | 1Inst
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