Adaptive refinement for arbitrary finite-element spaces with hierarchical bases
Journal of Computational and Applied Mathematics
Local bisection refinement for N-simplicial grids generated by reflection
SIAM Journal on Scientific Computing
A recursive approach to local mesh refinement in two and three dimensions
Journal of Computational and Applied Mathematics
Computing
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
The 4-triangles longest-side partition of triangles and linear refinement algorithms
Mathematics of Computation
A 3D refinement/derefinement algorithm for solving evolution problems
Applied Numerical Mathematics - Special issue on numerical grid generation-technologies for advanced simulations
Locally Adapted Tetrahedral Meshes Using Bisection
SIAM Journal on Scientific Computing
hp-Approximation Theory for BDFM and RT Finite Elements on Quadrilaterals
SIAM Journal on Numerical Analysis
Unified multilevel adaptive finite element methods for elliptic problems
Unified multilevel adaptive finite element methods for elliptic problems
Adaptive Finite Element Methods with convergence rates
Numerische Mathematik
On the Interpolation Error Estimates for $Q_1$ Quadrilateral Finite Elements
SIAM Journal on Numerical Analysis
Constraint-Free Adaptive FEMs on Quadrilateral Nonconforming Meshes
Journal of Scientific Computing
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Hanging nodes have some disadvantages in the implementation of adaptive finite element methods on quadrilateral meshes, which usually need further techniques to treat them. In this paper, we present a shape regular local refinement algorithm on quadrilateral meshes without hanging nodes, which can be viewed as an extension of the original red-green refinement proposed by Bank, Sherman, and Weiser [Refinement algorithms and data structures for regular local mesh refinement, in Scientific Computing: Applications of Mathematics and Computing to the Physical Sciences, R. S. Stepleman, ed., North-Holland, Amsterdam, 1983, pp. 3-17]. We also prove this quadrilateral red-green refinement can hold the shape regularity and conformity of quadrilateral meshes. Furthermore, we have successfully accomplished the adaptive finite element computation on quadrilateral meshes without hanging nodes. Numerical results show that the adaptive algorithm with this red-green refinement is quasi-optimal. Some properties of this local refinement on quadrilateral meshes are also covered in this paper.