Partial Differential Equation Toolbox

The Mesh Refiner

The PDE Toolbox is geared to elliptic problems. For reasons of accuracy and ill-conditioning, they require the elements not to deviate too much from being equilateral. Thus, even at essentially one-dimensional solution features, such as boundary layers, the refinement technique must guarantee reasonably shaped triangles.

When an element is refined, new nodes appear on its midsides, and if the neighbor triangle is not refined in a similar way, it is said to have hanging nodes. The final triangulation must have no hanging nodes, and they are removed by splitting neighbor triangles. To avoid further deterioration of triangle quality in successive generations, the "longest edge bisection" scheme Rosenberg-Stenger [8] is used, in which the longest side of a triangle is always split, whenever any of the sides have hanging nodes. This guarantees that no angle is ever smaller than half the smallest angle of the original triangulation.

Two selection criteria can be used. One, pdeadworst, refines all elements with value of the error indicator larger than half the worst of any element. The other, pdeadgsc, refines all elements with an indicator value exceeding a user-defined dimensionless tolerance. The comparison with the tolerance is properly scaled with respect to domain and solution size, etc.

The Termination Criteria

For smooth solutions, error equidistribution can be achieved by the pdeadgsc selection if the maximum number of elements is large enough. The pdeadworst adaption only terminates when the maximum number of elements has been exceeded. This mode is natural when the solution exhibits singularities. The error indicator of the elements next to the singularity may never vanish, regardless of element size, and equidistribution is too much to hope for.

The Error Indicator Function

Fast Solution of Poisson's Equation