98 research outputs found
Optimal Order Convergence Implies Numerical Smoothness
It is natural to expect the following loosely stated approximation principle
to hold: a numerical approximation solution should be in some sense as smooth
as its target exact solution in order to have optimal convergence. For
piecewise polynomials, that means we have to at least maintain numerical
smoothness in the interiors as well as across the interfaces of cells or
elements. In this paper we give clear definitions of numerical smoothness that
address the across-interface smoothness in terms of scaled jumps in derivatives
[9] and the interior numerical smoothness in terms of differences in derivative
values. Furthermore, we prove rigorously that the principle can be simply
stated as numerical smoothness is necessary for optimal order convergence. It
is valid on quasi-uniform meshes by triangles and quadrilaterals in two
dimensions and by tetrahedrons and hexahedrons in three dimensions. With this
validation we can justify, among other things, incorporation of this principle
in creating adaptive numerical approximation for the solution of PDEs or ODEs,
especially in designing proper smoothness indicators or detecting potential
non-convergence and instability
The maximum angle condition is not necessary for convergence of the finite element method
We show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems. This condition is only sufficient. In fact, finite element approximations may converge even though some dihedral angles of simplicial elements tend to π
On global and local mesh refinements by a generalized conforming bisection algorithm
We examine a generalized conforming bisection (GCB-)algorithm which allows both global and local nested refinements of the triangulations without generating hanging nodes. It is based on the notion of a mesh density function which prescribes where and how much to refine the mesh. Some regularity properties of generated sequences of refined triangulations are proved. Several numerical tests demonstrate the efficiency of the proposed bisection algorithm. It is also shown how to modify the GCB-algorithm in order to generate anisotropic meshes with high aspect ratios
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