Finite elements on dyadic grids with applications
The authors consider large-scale problems with a highly resolution scale in space and time. Using uniform orthogonal grids a mesh refinement in one region results in an accumulation of small cells in all coordinate directions even though generally the refinement is needed only in special regions. In order to reduce the computational costs finite element schemes with variable mesh sizes have been developed using triangular and tetrahedral subdivisions, Voronoi cell grids, deformed orthogonal grids and so on which fit the local accuracy requirements. Disadvantages of these methods are: complex algorithms and data structures have to be developed, unfavourable grid elements have to be avoided, and last not least, the conversion of data from and to regular orthogonal grid forms needed in software for industrial applications is difficult. Thus, the authors propose an alternative fixed-geometry hierarchical adaptive grid scheme of arbitrary dimension with strictly binary subdivisions, so-called dyadic grids, where a cell at level \(k\) is partitioned into two equal subcells of level \(k+1\). The application of the simple sequence of subdivisions to an initial grid of cubic cells results in a fixed-geometry scheme of rectangular meshes which are easier to represent than the above mentioned irregular grids. Polynomial splines \(P_c^g[G]\) with a continuity \(c \geq 0\) and a maximum degree \(g = 2c+1\) in each coordinate are defined over a dyadic grid \(G\), and a finite element basis for these dyadic splines is given including an algorithm for the construction of the basis. The method is demonstrated by applying it to least squares fitting of the two-dimensional Gaussian function, to the use of Galerkin's method with dyadic finite elements with linear splines \(P_0^1[G]\) to solve the two-dimensional heat diffusion equation, and to the biphase flow with moving fronts, a hard problem for irregular meshes.
scientific article; zbMATH DE number 5073219