Finite elements on dyadic grids with applications
type: Item
instance of: scholarly article
title: Finite elements on dyadic grids with applications
zbMATH Open document ID: 1106.65085
author: Cláudio G. S. Cardoso
author: Anamaria Gomide
author: Denis J. Schiozer
author: Jorge Stolfi
author: M. C. C. Cunha
published in: Mathematics and Computers in Simulation
publication date: 2006-11-15T00:00:00Z
review text: 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.
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 65M50
Mathematics Subject Classification ID: 76T30
Mathematics Subject Classification ID: 76M10
zbMATH DE Number: 5073219
zbMATH Keywords: dyadic grids
zbMATH Keywords: adaptive grids
zbMATH Keywords: dyadic splines
zbMATH Keywords: finite elements
zbMATH Keywords: multiphase flow
zbMATH Keywords: oil reservoir simulation
zbMATH Keywords: moving fronts
zbMATH Keywords: mesh refinement
zbMATH Keywords: large-scale problems
zbMATH Keywords: subdivisions
zbMATH Keywords: algorithms
zbMATH Keywords: least squares fitting
zbMATH Keywords: Galerkin's method
zbMATH Keywords: heat diffusion equation
MaRDI profile type: Publication
full work available at URL: j.matcom.2006.06.024
OpenAlex ID: W1965237402
cites work: Q4732025
cites work: Multi-Level Adaptive Solutions to Boundary-Value Problems
cites work: Mesh refinements for parabolic equations
cites work: Q3779695
cites work: Adaptive local grid refinement and multi-grid in numerical reservoir simulation
cites work: Propagation through numerical mesh refinement for hyperbolic equations
DOI: 10.1016/J.MATCOM.2006.06.024
description: scientific article; zbMATH DE number 5073219