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On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation. (English) Zbl 1065.65135
This interesting and useful paper is devoted to the theory of finite element methods dealing with planar quadrilateral and three-dimensional “hexahedral” elements (cells of the grid). These quasicube cells are the images of the reference cube $$[-1,1]^n,\;n=2,3$$ under $$n$$-linear transformations that can be determined by the images of vertices (in the given order). It is important that the images of edges are straight line segments, for $$n=3$$ the images of faces can be nonplanar. If $$n=2$$ then a quasicube is a strictly convex quadrilateral.
The author defines a measure of nonregularity of the cell that corresponds to its distinction from a parallelogram or a parallelepiped. He proves that under proper refinement procedures the introduced measure tends to zero. It enables him to replace $$n$$-linear transformations by affine ones and deal only with piecewise constant approximations for Jacobians. So the computational work for getting the corresponding linear system is essentially simplified without noticeable loss of the accuracy of the modified grid method. Numerical examples are given.

##### MSC:
 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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