scholarly journals Adaptive finite element-boundary element solution of boundary value problems

1999 ◽  
Vol 106 (2) ◽  
pp. 307-316 ◽  
Author(s):  
O. Steinbach
Keyword(s):  
Finite Element ◽  
Boundary Value Problems ◽  
Boundary Element ◽  
Boundary Value ◽  
Adaptive Finite Element ◽  
Element Solution ◽  
Element Boundary

scholarly journals H-Adaptive Finite Element Methods for 1D Stationary High Gradient Boundary Value Problems

2021 ◽  
Vol 8 (6) ◽  
pp. 967-973
Author(s):  
Collins Olusola Akeremale ◽  
Oluwasegun Adeyemi Olaiju ◽  
Su Hoe Yeak
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
High Gradient ◽  
Adaptive Fem ◽  
Element Method

This article considered the traditional finite element method (FEM) and adaptive finite element method (FEM) for the numerical solution of the one-dimensional boundary value problems. We established the preference or the superiority of the h-adaptive FEM to traditional FEM in high gradient problems in terms of accuracy and cost of computation. Numerical examples which confirm the performance and adaptability of the h-adaptive method over the traditional finite element method and the high accuracy of the numerical solution are presented. Detailed error analysis of linear elements was also discussed. In conclusion, h-adaptive FEM is recommended for complex systems with high gradient problems.

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.

Sign in / Sign up

Export Citation Format

Share Document