A novel approach to anisotropic mesh-refinement of linear triangular elements in the finite element method is proposed. Using the method of solving a hierarchically refined dual problem in the a posteriori error analysis, indicators for edge refinement, rather than element refinement, can be obtained. A complete adaptive strategy is presented and illustrated by some numerical examples.

Designing the optimal finite element mesh for a certain problem is a key issue in order to make inexpensive and accurate computations. Standard

Developments in establishing strategies for goal-oriented error computations rely heavily on the idea of solving a dual problem, confer to [

Introducing an enhanced dual solution in terms of hierarchically added basis, as introduced in [

In this paper we first recall the general framework for a posteriori error computation for an abstract problem. Secondly, we discuss the relations between the a posteriori error estimation and error indicators related to the discretization. Next, a strategy for subdividing the mesh for given error indicators is suggested. Finally, we study the procedure on two test problems on a unit square, comparing anisotropic mesh-refinement with a (nearly) isotropic strategy. The Poisson problem illustrates the behavior for a problem of isotropic nature, while we use a reaction-diffusion problem with low conductivity to induce strong anisotropy along boundaries.

For completeness, we recall the general strategy in establishing the error representation for an abstract problem as described by for example, Eriksson et al. [

In order to define the corresponding finite element solution, we introduce the approximation space

We will now introduce the appropriate secant form of

From (

We will now turn to the error analysis. In order to retain maximal generality in the formulation, we select the appropriate goal-oriented error measure

We are now in the position to introduce the variational format of the dual problem. Find

Having computed

In general,

In the case of a linear problem, the secant form

The exact error representation is given by (

First, we note that the dual problem can be solved using the same FE-mesh as for the primal problem, that is,

Based on the error estimation above, we wish to use an adaptive

Usually, error indicators are obtained through restricting the error estimator (

We now propose, instead of computing the contributions of error from each element, to derive the error estimate as a sum of contributions from each element edge

First, we write the error contributions as a sum over the added basis functions. The enhancement space

We now turn to the case of using piecewise linear approximation for the primal problem (i.e.,

The conventional linear basis functions on a triangular element in 2D used for the FE-approximation.

Hierarchical basis functions utilized in error analysis; 3 basis functions constituting the complete quadratic polynomial space (top) and 1 (parasitic) basis function of cubic order (bottom).

Illustration of the similarity in adding a quadratic basis function (a) and subdividing the element with remaining linear approximation (b).

In view of the similarity of the quadratic basis function and the piecewise linear basis function that corresponds to a subdivision of that edge, confer to Figure

Concerning the internal bubble function, that has its support inside an element, we distribute its error contribution over the surrounding edges based on fractions of the circumference. Thus the coefficients

In our experience, the use of a fully quadratic basis (one order higher in convergence) is in many cases sufficient to obtain sharp error estimate, confer to [

In the case of element indicators, one important issue is to control the

In the proposed method, the refinement of an element is governed by which edge/edges of the element is marked for division due to high error contribution. In this fashion, we wish to construct the optimal mesh, depending on the problem, with respect to both size, aspect ratio, and orientation of elements. If the nature of the error (generation and transport) is anisotropic, the algorithm will generate an anisotropic mesh, while for an isotropic behavior of the error, an isotropic mesh will develop during remeshing.

Turning again to the case of linear triangular elements, the refinement of an element is based on the edges to be subdivided, 0, 1, 2, or 3. In the case that 1 edge is indicated for subdivision, two new elements are created. If all three edges are marked for subdivision, an isotropic refinement of the old element into 4 new elements is performed. In the case that 2 of the edges are indicated, the old triangle is subdivided into three new triangles. Two possible subdivisions are possible, based on which of the two error indicators (on marked edges) is the highest. The edge corresponding to the highest error indicator is made vertex for all three triangles, thus enriching interpolation quality in the element primarily along that edge. The possible subdivisions of a triangle are illustrated in Figure

Local subdivision of triangular elements with given edges to refine. 1, 2, or 3 edges can be marked for subdivision (indicated with circles,

Note that using the proposed method, each element is refined independently of its neighbors once the subdivided edges are indicated on a

The adaptive refinement is set to perform mesh-refinements until a certain tolerance is met, that is, until

Considering the linear triangular elements, we map the basis functions from a reference element onto the actual element in

We will investigate two scalar problems defined in 2D on the unit square,

We study the model problem

Solution for the Poisson problem, illustrated by the isolines of the solution.

Convergence of error with refinement (Ndof = no. of degrees of freedom) using different mesh-adaption strategies for the Poisson problem.

Final meshes using different refinement strategies for the Poisson problem. 14th isotropic mesh with

We study the model problem

Solution for the reaction-diffusion problem, illustrated by the isolines for the solution. Note the pronounced boundary layers produced by the Dirichlet boundary conditions.

Convergence of error with refinement (Ndof = no. of degrees of freedom) using different mesh-adaption strategies for the reaction-diffusion problem.

Final meshes using different refinement strategies for the reaction-diffusion problem. 15th isotropic mesh with

The proposed method shows promising behavior in the following respects: (i) it is more efficient than an isotropic refinement strategy for problems of anisotropic nature, (ii) its behavior is similar to that of the isotropic strategy when the nature of the problem is isotropic. Thus the method is truly adaptive, in the sense that the mesh becomes anisotropic if and only if, the nature of the error demands it, without adding any extra cost to the computation. Furthermore, the refinement procedure is extremely simple, since all element subdivisions are completely independent of their neighbors once the edges to subdivide are indicated.

For future work, the method seems appealing to extend to 3D due to (i) the efficiency in terms of degrees of freedom and (ii) the ease of implementation. If a corresponding method is to be used for quadrilateral (or hexahedral) elements, additional constraints must be applied, in order to control the legality of the iso-parametric mapping.