A fast algorithm for point-location in a finite element mesh

An algorithm for the point-location problem in 2D finite element meshes as a special case of plane straight-line graphs (PSLG) is presented. The element containing a given point P is determined combining a quadtree data structure to generate a quaternary search tree and a local search wave using adjacency information. The preprocessing construction of the search tree has a complexity of O(n · log(n)) and requires only pointer swap operations. The query time to locate a start element for local search is O(log(n)) and the final point search by 'point-in-polygon' tests is independent of the total number of elements in the mesh and thus determined in constant time. Although the theoretical efficiency estimates are only given for quasi-uniform meshes, it is shown in numerical examples, that the algorithm performs equally well for meshes with extreme local refinement.