Journal ArticleDOI

# Precise FEM solution of a corner singularity using an adjusted mesh

10 Apr 2005--Vol. 47, pp 1285-1292
TL;DR: In this article, the authors present an alternative approach for flow problems on domains with corner singularities, using the a priori error estimates and asymptotic expansion of the solution to derive an algorithm for refining the mesh near the corner.
Abstract: Within the framework of the finite element method problems with corner-like singularities (e.g. on the well-known L-shaped domain) are most often solved by the adaptive strategy: the mesh near the corners is refined according to the a posteriori error estimates. In this paper we present an alternative approach. For flow problems on domains with corner singularities we use the a priori error estimates and asymptotic expansion of the solution to derive an algorithm for refining the mesh near the corner. It gives very precise solution in a cheap way. We present some numerical results. Copyright © 2005 John Wiley & Sons, Ltd.

### Introduction

• The local behaviour of the solution near the singular point is used to design a mesh which is adjusted to the shape of the solution.
• Then the authors use this adjusted mesh for the numerical solution of flow in the channel with corners.

### Model problem

• The authors consider two-dimensional flow of viscous, incompressible fluid described by NavierStokes equations in a domain with corner singularity, cf. Fig.
• Due to symmetry, the authors solve the problem only on the upper half of the channel.

### Algorithm derivation

• Similar results have been proved for the Navier-Stokes equations.
• Using this algorithm the authors obtained satisfactory results.

### Evaluation of the error

• Here n means the number of all elements in the domain.
• More about the evaluation of error in [2].

### Numerical results

• On Figures 5-8 the authors present the graphical output of entities that chartacterize the flow in the channel.
• Figure 6 with contours of velocity vy shows that the solution is satisfactory smooth on refined area.
• On Figs. 7-8 the authors observe how strong the singularity is, both for velocity and pressure (note that here the flow is from the right to the left, to have better view).
• Fig. 6: Isolines of velocity vy Fig. 7: Velocity component vy Fig. 8: Pressure near the corner On Figs. 9-11 the authors show the errors on elements.

### Conclusions

• Presented results give satisfactory confirmation of developed algorithm.
• Application of a priori error estimates of finite element method for mesh refinement near singularity is very efficient for their problem what can be seen especially from obtained errors on elements which are very uniformly distributed.
• This approach is an alternative to ’classical’ one using adaptive mesh refinement, which is still much more robust.

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