### Adaptavity for the Control Volume Finite Element Method in Convection-Diffusion Problems

#### Abstract

In Computational Fluid Dynamics it is usual to find the problem of increasing the accuracy of a solution without adding unnecessary degrees of freedom. It is therefore necessary to update the mesh

so as to ensure that it becomes fine enough in the critical region while remaining reasonably coarse in the rest of the domain. Local a posteriori error estimators are the adequate tool for identifying automatically these critical regions. They should use only given data and the numerical solution itself.

In this work the Control Volume Finite Element Method (CVFEM) for the Conveetion:Diffusion equation is considered. This is a nonconforming method in the sense that the interpolant space for the solution is not a subset of Hl. Despite of this fact, many years of numerical experiences have established the excellent behaviour of this method in non-selfadjoint problems.

In the conforming case several approaches have been introduced for selfadjoint problems by using the residual equations. In order to extend these techniques to the case we are dealing with, we have considered the treatment of the consistency terms arising in the error equation and the convective term wich is the no-selfadjoint part of the problem. Although some a posteriori error estimators for this problem have already been presented in the literature, most of them lack rigorous mathematical proof.

We present an error estimator that is a global upper bound of the true error under some hypotheses [11]. It has been included in a CVFEM code of our own. This code has been coupled together with an automatic mesh generator in order to obtain an adaptive loop. Evidence of the adequate behaviour of the adap ive procedure is given through numerical experimentation in well-known

benchmark problems.

so as to ensure that it becomes fine enough in the critical region while remaining reasonably coarse in the rest of the domain. Local a posteriori error estimators are the adequate tool for identifying automatically these critical regions. They should use only given data and the numerical solution itself.

In this work the Control Volume Finite Element Method (CVFEM) for the Conveetion:Diffusion equation is considered. This is a nonconforming method in the sense that the interpolant space for the solution is not a subset of Hl. Despite of this fact, many years of numerical experiences have established the excellent behaviour of this method in non-selfadjoint problems.

In the conforming case several approaches have been introduced for selfadjoint problems by using the residual equations. In order to extend these techniques to the case we are dealing with, we have considered the treatment of the consistency terms arising in the error equation and the convective term wich is the no-selfadjoint part of the problem. Although some a posteriori error estimators for this problem have already been presented in the literature, most of them lack rigorous mathematical proof.

We present an error estimator that is a global upper bound of the true error under some hypotheses [11]. It has been included in a CVFEM code of our own. This code has been coupled together with an automatic mesh generator in order to obtain an adaptive loop. Evidence of the adequate behaviour of the adap ive procedure is given through numerical experimentation in well-known

benchmark problems.

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Güemes 3450

S3000GLN Santa Fe, Argentina

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Fax: 54-342-4511169

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**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**