scholarly journals An adaptive algorithm for multivariate approximation giving optimal convergence rates

1979 ◽  
Vol 25 (4) ◽  
pp. 337-359 ◽  
Author(s):  
Carl de Boor ◽  
John R. Rice
Keyword(s):  
Adaptive Algorithm ◽  
Convergence Rates ◽  
Multivariate Approximation ◽  
Optimal Convergence ◽  
Optimal Convergence Rates

Mixed Schemes for Fourth-Order DIV Equations

2019 ◽  
Vol 19 (2) ◽  
pp. 341-357
Author(s):  
Ronghong Fan ◽  
Yanru Liu ◽  
Shuo Zhang
Keyword(s):  
Theoretical Analysis ◽  
Fourth Order ◽  
Convergence Rates ◽  
Mixed Formulation ◽  
Optimal Convergence ◽  
Mixed Formulations ◽  
Optimal Convergence Rates

AbstractIn this paper, stable mixed formulations are designed and analyzed for the quad div problems under two frameworks presented in [23] and [22], respectively. Analogue discretizations are given with respect to the mixed formulation, and optimal convergence rates are observed, which confirm the theoretical analysis.

scholarly journals A New Generalization of the P1 Non-Conforming FEM to Higher Polynomial Degrees

2017 ◽  
Vol 17 (1) ◽  
pp. 161-185 ◽  
Author(s):  
Mira Schedensack
Keyword(s):  
Adaptive Algorithm ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Projection Property ◽  
Polynomial Degree ◽  
Poisson Problem ◽  
Optimal Convergence Rates ◽  
New Formulation ◽  
Three Space

AbstractThis paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

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