Partial Differential Equation Toolbox

Adaptive Mesh Refinement

The toolbox has a function for global, uniform mesh refinement. It divides each triangle into four similar triangles by creating new corners at the midsides, adjusting for curved boundaries. You can assess the accuracy of the numerical solution by comparing results from a sequence of successively refined meshes. If the solution is smooth enough, more accurate results may be obtained by extrapolation.

The solutions of the toolbox equation often have geometric features like localized strong gradients. An example of engineering importance in elasticity is the stress concentration occurring at reentrant corners such as the MATLAB favorite, the L-shaped membrane. Then it is more economical to refine the mesh selectively, i.e., only where it is needed. When the selection is based on estimates of errors in the computed solutions, a posteriori estimates, we speak of adaptive mesh refinement. See adaptmesh for an example of the computational savings where global refinement needs more than 6000 elements to compete with an adaptively refined mesh of 500 elements.

The adaptive refinement generates a sequence of solutions on successively finer meshes, at each stage selecting and refining those elements that are judged to contribute most to the error. The process is terminated when the maximum number of elements is exceeded or when each triangle contributes less than a preset tolerance. You need to provide an initial mesh, and choose selection and termination criteria parameters. The initial mesh can be produced by the initmesh function. The three components of the algorithm are the error indicator function, which computes an estimate of the element error contribution, the mesh refiner, which selects and subdivides elements, and the termination criteria.

Nonlinear Equations

The Error Indicator Function