Solvability of one nonlocal Dirichlet problem for the Poisson equation

2019 ◽  
Vol 50 (1) ◽  
pp. 67-88
Author(s):  
Batirkhan Turmetov ◽  
Valery Karachik
Keyword(s):  
Dirichlet Problem ◽  
Poisson Equation ◽  
The Poisson Equation

scholarly journals A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square

2019 ◽  
Vol 53 (3) ◽  
pp. 987-1003 ◽  
Author(s):  
Claudio Canuto ◽  
Ricardo H. Nochetto ◽  
Rob P. Stevenson ◽  
Marco Verani
Keyword(s):  
Dirichlet Problem ◽  
Poisson Equation ◽  
Galerkin Approximation ◽  
Error Reduction ◽  
Two Dimensional ◽  
Dirichlet Problems ◽  
Polynomial Degree ◽  
Unit Square ◽  
The Poisson Equation ◽  
Dimensional Unit

Both practice and analysis of p-FEMs and adaptive hp-FEMs raise the question what increment in the current polynomial degree p guarantees a p-independent reduction of the Galerkin error. We answer this question for the p-FEM in the simplified context of homogeneous Dirichlet problems for the Poisson equation in the two dimensional unit square with polynomial data of degree p. We show that an increment proportional to p yields a p-robust error reduction and provide computational evidence that a constant increment does not.

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