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Publication# An adaptive algorithm for the transport equation with time dependent velocity

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.

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Related concepts (20)

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

Time

Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component

Space

Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it,

Related publications (26)

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We developed a space and time adaptive method to simulate electroosmosis and mass transport of a sample concentration within a network of microchannels. The space adaptive criteria is based on an error estimator derived using anisotropic interpolation estimates and a post-processing procedure. In order to improve the accuracy of the numerical solution and to reduce even further the computational cost of the numerical simulation, a time adaptive procedure is combined with the one in space. To do so, a time error estimator is derived for a first model problem, the linear heat equation discretized in time with the Crank-Nicolson scheme. The main difficulty is then to obtain an optimal second order error estimator. Applying standard energy techniques with a continuous, piecewise linear approximation in time fail in recovering the optimal order. To restore the appropriate rate of convergence, a continuous piecewise quadratic polynomial function in time is needed. For this purpose, two different quadratic functions are introduced and two different time error estimators are then derived. It turns out that the second error estimator is more efficient than the first one when considering our adaptive algorithm. Thus, using the second quadratic polynomial, an upper bound for the error is derived for a second model problem, the time-dependent convection-diffusion problem discretized in time with the Crank-Nicolson scheme. The corresponding space and time error estimators are finally used for the numerical simulation of mass transport of a sample concentration within a complex network of microchannels driven by an electroosmotic flow and/or by a pressure-driven flow. Numerical results presented show the efficiency and the robustness of this approach.

Marco Picasso, Virabouth Prachittham

In this paper we derive two a posteriori upper bounds for the heat equation. A continuous, piecewise linear finite element discretization in space and the Crank-Nicolson method for the time discretization are used. The error due to the space discretization is derived using anisotropic interpolation estimates and a postprocessing procedure. The error due to the time discretization is obtained using two different continuous, piecewise quadratic time reconstructions. The first reconstruction is developed following G. Akrivis, C. Makridakis, and R. H. Nochetto [Math. Comp., 75 (2006), pp. 511-531], while the second one is new. Moreover, in the case of isotropic meshes only, upper and lower bounds are provided as in [R. Verfurth, Calcolo, 40 (2003), pp. 195 212]. An adaptive algorithm is developed. Numerical studies are reported for several test cases and show that the second error estimator is more efficient than the first one. In particular, the second error indicator is of optimal order with respect to both the mesh size and the time step when using our adaptive algorithm.

2009Mathematical and numerical aspects of viscoelastic flows are investigated here. Two simplified mathematical models are considered. They are motivated by a splitting algorithm for solving viscoelastic flows with free surfaces. The first model is a simplified Oldroyd-B model. Existence on a fixed time interval is proved in several Banach spaces provided the data are small enough. Short time existence is also proved for arbitrarily large data in Hölder spaces for the time variable. These results are based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework. Then, the extension of these results to a stochastic simplified Hookean dumbbells model is discussed. Because of the presence of the Brownian motion, existence in a fixed time interval, provided the data are small enough, is proved only in some of the Banach spaces considered previously. The dumbbells' elongation is split in two parts, one satisfying a standart stochastic differential equation, the other satisfying a partial differential equation with a stochastic source term. A finite element discretization in space is also proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates. A numerical algorithm for solving viscoelastic flows with free surfaces is also described. This algorithm is based on a splitting method in time and two different meshes are used for the space discretization. Convergence of the numerical model is checked for the pure extensional flow and the filling of a pipe. Then, numerical results are reported for the stretching of a filament and for jet buckling.