Feedback and adaptive finite element solution of one-dimensional boundary value problems

1984 ◽  
Vol 44 (1) ◽  
pp. 75-102 ◽  
Author(s):  
I. Bubuška ◽  
M. Vogelius
Keyword(s):  
Finite Element ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Finite Element Solution ◽  
Adaptive Finite Element ◽  
Element Solution ◽  
One Dimensional ◽  
Dimensional Boundary

Maximum norm error estimates for Neumann boundary value problems on graded meshes

2018 ◽  
Vol 40 (1) ◽  
pp. 474-497 ◽  
Author(s):  
Thomas Apel ◽  
Sergejs Rogovs ◽  
Johannes Pfefferer ◽  
Max Winkler
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Error Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Neumann Boundary ◽  
Finite Element Solution ◽  
Element Solution ◽  
Polygonal Domains

AbstractThis paper deals with a priori pointwise error estimates for the finite element solution of boundary value problems with Neumann boundary conditions in polygonal domains. Due to the corners of the domain, the convergence rate of the numerical solutions can be lower than in the case of smooth domains. As a remedy, the use of local mesh refinement near the corners is considered. In order to prove quasi-optimal a priori error estimates, regularity results in weighted Sobolev spaces are exploited. This is the first work on the Neumann boundary value problem where both the regularity of the data is exactly specified and the sharp convergence order $h^{2} \lvert \ln h \rvert $ in the case of piecewise linear finite element approximations is obtained. As an extension we show the same rate for the approximate solution of a semilinear boundary value problem. The proof relies in this case on the supercloseness between the Ritz projection to the continuous solution and the finite element solution.

Perturbation finite element solution for chemo-elastic boundary value problems under chemical equilibrium

2019 ◽  
Vol 35 (5) ◽  
pp. 981-991 ◽  
Author(s):  
Peter L. Bishay ◽  
Jan Sladek ◽  
Nicholas Fabry ◽  
Vladimir Sladek ◽  
Chuanzeng Zhang
Keyword(s):  
Finite Element ◽  
Boundary Value Problems ◽  
Chemical Equilibrium ◽  
Boundary Value ◽  
Finite Element Solution ◽  
Element Solution ◽  
Elastic Boundary
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