Solutions of partial differential equations may form singularity or geometric structures may develop in a priori unknown locations. When using mesh based methods (e.g., finite element or finite volume methods) for the approximation of such solutions, adaptive mesh refinement allows for efficient computations. The main idea is to refine only in locations where the error is expected to be large compared to the overall error.
Depending on the specific numerical scheme there are different types of meshes and adaptive refinement schemes. We investigate their properties and develop theoretical tools such as projection operators to analyse methods on adaptively refined meshes. When applied in conjunction with a posteriori error estimates adaptive mesh refinement allows for optimal convergence results for some partial differential equations.
