A posteriori pointwise error computation for 2-D transport equations based on the variational multiscale method

Abstract

This article presents a general framework to estimate the pointwise error of linear partial differential equations. The error estimator is based on the variational multiscale theory, in which the error is decomposed in two components according to the nature of the residuals: element interior residuals and interelement jumps. The relationship between the residuals (coarse scales) and the error components (fine scales) is established, yielding to a very simple model. In particular, the pointwise error is modeled as a linear combination of bubble functions and Green's functions. If residual-free bubbles and the classical Green's function are employed, the technology leads to an exact explicit method for the pointwise error. If bubble functions and free-space Green's functions are employed, then a local projection problem must be solved within each element and a global boundary integral equation must be solved on the domain boundary. As a consequence, this gives a model for the so-called fine-scale Green's functions. The numerical error is studied for the standard Galerkin and SUPG methods with application to the heat equation, the reaction-diffusion equation and the convection-diffusion equation. Numerical results show that stabilized methods minimize the propagation of pollution errors, which stay mostly locally.

Variational Multiscale error estimator for anisotropic adaptive fluid mechanic simulations: Application to convection–diffusion problems

“…Approximating u bub (x) by Taylor series and neglecting the second order terms, we have an expression of u bub (x) close to the centroid c i of the element [35]: As said before, we make the assumption that the residual f − Lu h is P0. Therefore, we have:…”

“…In this section, we use the work of Irisarri et al in [35]. This time, we apply a pointwise computation of the error estimator.…”

“…Practical aspects of computation As proposed by Irisarri et al in [35], the way of obtaining a numerical solution of u bub is by solving the following discrete problem: (25) This time, the error component is expressed with a combination of bubble functions:…”

“…The error estimator is computed thanks to the pointwise error estimation of Irisarri et al in [35]. It uses a set of bubble functions as a substitution of the subscales Green's functions.…”

“…Referring to Irisarri et al, the second method uses a linear combination of bubble functions to establish a pointwise computation of the error. It has been developed for one-dimensional transport equation in [34] and for 2D transport equation in [35].…”