L 2 error estimates for finite elements with interpolated boundary conditions

1973 ◽  
Vol 21 (4) ◽  
pp. 345-349 ◽  
Author(s):  
Alan E. Berger
Keyword(s):  
Finite Elements ◽  
Boundary Conditions ◽  
Error Estimates

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

2020 ◽  
Vol 20 (2) ◽  
pp. 361-378
Author(s):  
Tamal Pramanick ◽  
Rajen Kumar Sinha
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Error Estimates ◽  
Finite Element Approximation ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Dirichlet Problems ◽  
Element Mesh ◽  
Polygonal Domains ◽  
Numer Math

AbstractThe purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the {L^{\infty}(H^{1})}-norm and {L^{\infty}(L^{2})}-norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

scholarly journals On Superconvergence of a Gradient for Finite Element Methods for an Elliptic Equation with the Nonsmooth Right–hand Side

2002 ◽  
Vol 2 (3) ◽  
pp. 295-321 ◽  
Author(s):  
Alexander Zlotnik
Keyword(s):  
Finite Elements ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Boundary Function ◽  
Right Hand ◽  
Priori Error Estimates ◽  
The Right ◽  
2D And 3D

AbstractThe elliptic equation under the nonhomogeneous Dirichlet boundary condition in 2D and 3D cases is solved. A rectangular nonuniform partition of a domain and polylinear finite elements are taken. For the interpolant of the exact solution u, a priori error estimates are proved provided that u possesses a weakened smoothness. Next error estimates are in terms of data. An estimate is established for the right–hand side f of the equation having a generalized smoothness. Error estimates are derived in the case of f which is not compatible with the boundary function. The proofs are based on some propositions from the theory of functions. The corresponding lower error estimates are also included; they justify the sharpness of the estimates without the logarithmic multipliers. Finally, we prove similar results in the case of 2D linear finite elements and a uniform partition.

Prediction of Rail Profile Wear with the Increase of Vehicle Speed

2013 ◽  
Vol 716 ◽  
pp. 659-662
Author(s):  
Dong Hyong Lee ◽  
Jeong Won Seo ◽  
Seok Jin Kwon ◽  
Ha Young Choi
Keyword(s):  
Numerical Analysis ◽  
Finite Elements ◽  
Boundary Conditions ◽  
Rolling Contact ◽  
Finite Elements Method ◽  
Vehicle Speed ◽  
Two Dimensional ◽  
Contact Wear ◽  
Rail Wear ◽  
Finite Elements Model

A method to simulate rolling contact wear in a rail surface was developed using the finite elements method and numerical analysis. A two-dimensional finite elements model was used in order to reduce the calculation time and boundary conditions to prevent excessive deformation of a wheel and a rail were applied. A numerical analysis of rail wear at rolling contact was predicted using the Archards equation. In addition, the characteristics of rail wear with the increasing speed of vehicle were analyzed. Results show that there was not a large difference in the depths of wear on the rail head with increasing vehicle speed, but the wear on the rail gauge corner increased with increasing vehicle speed.

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