Meshverfijning

Meshverfijning, also known as mesh refinement, refers to techniques for increasing the resolution of a computational mesh in regions where higher accuracy is required. It is a key component of adaptive methods, often called adaptive mesh refinement (AMR). In numerical simulations of partial differential equations, the solution is approximated on a mesh; refining the mesh by subdividing elements yields more degrees of freedom and typically improves accuracy, especially near sharp gradients, boundary layers, or singularities.

Refinement can be classified by how the mesh is altered: h-refinement reduces element size; p-refinement increases the polynomial order of the basis functions within elements; r-refinement relocates nodes to improve approximation while keeping the total number of elements roughly constant. Adaptive mesh refinement uses error indicators or a posteriori error estimators to decide where and how to refine. Common strategies include marking elements where the estimated error is large and applying targeted refinement, possibly with coarsening in other regions to maintain balance.

Refinement operations are implemented in various mesh types, including structured and unstructured meshes. In 2D, longest-edge bisection is a standard approach; in 3D, octree or other hierarchical schemes are common. AMR is widely used in finite element and finite volume methods across engineering fields, such as computational fluid dynamics, structural mechanics, electromagnetics, and geophysics, to resolve features like vortices, shock fronts, or boundary layers more accurately without a uniform global mesh refinement.

Benefits include higher accuracy where needed and reduced overall computational cost compared with a uniformly refined mesh. Challenges involve maintaining mesh quality, handling nonconforming elements and hanging nodes, developing reliable error estimators, and managing load balancing and data structures in parallel simulations.