An adaptive stabilized finite element scheme for a water quality model

Abstract

Residual type a posteriori error estimators are introduced in this paper for an advection-diffusion-reaction problem with a Dirac delta source term. The error is measured in an adequately weighted W 1,p -norm. These estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.

A posteriorierror estimates for elliptic problems with Dirac measure terms in weighted spaces

“…The main goal of this article is to develop a posteriori error estimates for elliptic second order partial differential equations on two-and three-dimensional domains with point sources. Elliptic problems with Dirac measure source terms arise in modeling different applications as, for instance, the electric field generated by a point charge, the acoustic monopoles or pollutant transport and degradation in an aquatic media where, due to the different scales involved, the pollution source is modeled as supported on a single point [3]. Other applications involve the coupling between reaction-diffusion problems taking place in domains of different dimension, which arise in tissue perfusion models [11].…”

“…A posteriori error estimates on two dimensional domains have been obtained by Araya et al [3,4] for the error measured in L p (1 < p < ∞) and W 1,p (p 0 < p < 2) for certain value of p 0 , and by Gaspoz et al [16] for the error measured in H s (1/2 < s < 1). Recall that the usual test and ansatz space for elliptic problems is the Sobolev space H 1 0 = W 1,2 0 .…”

“…Point sources do not belong to the dual space of H 1 0 , because H 1 0 is not immersed into the space of continuous functions, but in two dimensions very little is missing, since functions in W 1,p 0 and H s 0 are continuous if p > 2 and s > 1. This fact was exploited in [3,4] and [16].…”

Depth‐Dependent Geometry of the Liquiñe‐Ofqui Fault Zone and Its Relation to Paths of Slab‐Derived Fluids

“…A.) (Araya and Venegas, 2014;Araya et al, 2007bAraya et al, , 2007a, which solves the steady state heat equation. The model consists of an oceanic slab with constant dip angle, subducting beneath a nondeforming continental plate.…”

Adaptive control of local errors for elliptic problems using weighted Sobolev norms

“…We consider both the case of regular sources, i.e., , and the case of a singular point source assuming that , where and is the Dirac delta distribution supported at an inner point x 0 of Ω. Applications arise in different areas, such as in the study of pollutant diffusion in aquatic media , in the mathematical modeling of electromagnetic fields , or in optimal control of elliptic problems with state constraints . Other applications involve the coupling between reaction‐diffusion problems taking place in domains of different dimension, which arise in tissue perfusion models .…”