A fews snags in mesh adaptation loops

Abstract

The first stage in an adaptive finite element scheme (cf. [CAS95, bor1]) consists in creating an initial mesh of a given domain Ω, which is used to perform an initial computation (for example a flow solver). A size specification field is deduced (e.g. at the vicinity of each mesh vertex, the desired mesh size is specified), based on the numerical results. If the mesh does not satisfy the size specification field, then a new constrained mesh, governed by this field, is constructed. The size specification field is usually obtained via an error estimate [FOR, VER96]. Actually, the estimation gives a discrete size specification field. Using an adequate size interpolation over the mesh elements, a continuous field is then obtained.

Metrics are commonly used to normalize the mesh size specification to one in any direction (cf. [VAL92]), and are defined as a symmetric positive definite matrix associated to any point of the domain.

A classical adaptation loop is: 0 Build a initial mesh $\mathcal{T}_h^0$ 1 loop $i$ = 0, ...

• • Solve your problem on mesh $\mathcal{T}_h^i$

• • Compute an error indicator , and if the error is small enough then stop.

• • Compute a metric $\mathcal{M}^{i + 1}$,

• • Bound, regularize the metric $\mathcal{M}^{i + 1}$,

• • Compute a new unit mesh $\mathcal{T}_h^{i + 1}$ with respect to the new metric.

In this kind of algorithm, there are two problematic cases:

• One) if the minimal mesh size is reached then we generally lose the anisotropy of the mesh in this region.

• Two) In the adaptation loop, we use a hidden scheme to evaluate the metric, so some-times the mesh size to compute a good approximation of the solution is incompatible with the scheme to get a good approximation of the metric.

First, we do the numerical experiment to show this two snags. All the experiments are done with FreeFem++ software , see [freefempp, DAN03].

In this article we present the classical mesh adaptation with metric in section 2. And in section 3 we present the first trouble and some way to solve it. In section 4, a second problem is described and we explain when it occurs.