Finite element approximation with numerical integration for differential eigenvalue problems

2015 ◽  
Vol 93 ◽  
pp. 206-214 ◽  
Author(s):  
Sergey I. Solov'ëv
Keyword(s):  
Finite Element ◽  
Numerical Integration ◽  
Eigenvalue Problems ◽  
Finite Element Approximation ◽  
Element Approximation

scholarly journals Regularity and hp discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials

2019 ◽  
Vol 29 (08) ◽  
pp. 1585-1617 ◽  
Author(s):  
Yvon Maday ◽  
Carlo Marcati
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Sobolev Spaces ◽  
Eigenvalue Problems ◽  
Weighted Sobolev Spaces ◽  
Finite Element Approximation ◽  
Three Dimensions ◽  
Element Approximation ◽  
Singular Potentials ◽  
Dg Method

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [Formula: see text] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [Formula: see text] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.

Finite element approximation of eigenvalue problems

Acta Numerica ◽  
2010 ◽  
Vol 19 ◽  
pp. 1-120 ◽  
Author(s):  
Daniele Boffi
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Differential Forms ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Galerkin Approximations ◽  
Laplace Eigenvalue ◽  
Main Emphasis ◽  
Adjoint Operators

We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.

scholarly journals The effect of numerical integration in mixed finite element approximation in the simulation of miscible displacement

2017 ◽  
Vol 6 (2) ◽  
pp. 44
Author(s):  
Pongui ngoma Diogene Vianney ◽  
Nguimbi Germain ◽  
Likibi Pellat Rhoss Beaunheur
Keyword(s):  
Finite Element ◽  
Numerical Integration ◽  
Mixed Finite Element ◽  
Sufficient Conditions ◽  
Finite Element Approximation ◽  
Miscible Displacement ◽  
Optimal Order ◽  
Element Approximation ◽  
Quadrature Scheme ◽  
Effect Of Numerical Integration

We consider the effect of numerical integration in finite element  procedures applied to a nonlinear system of two coupled partial differential equations describing the miscible displacement of one incompressible fluid by another in a porous meduim. We consider the use of the numerical quadrature scheme for approximating the pressure and velocity by a mixed method using Raviart - Thomas space of index  and the concentration by a standard Galerkin method. We also give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order estimates are derived when the imposed external flows are smoothly distributed.

scholarly journals A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems

2003 ◽  
Vol 13 (08) ◽  
pp. 1219-1229 ◽  
Author(s):  
Ricardo G. Durán ◽  
Claudio Padra ◽  
Rodolfo Rodríguez
Keyword(s):  
Finite Element ◽  
Eigenvalue Problems ◽  
Finite Element Approximation ◽  
Posteriori Error ◽  
Higher Order ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Higher Order Terms

This paper deals with a posteriori error estimators for the linear finite element approximation of second-order elliptic eigenvalue problems in two or three dimensions. First, we give a simple proof of the equivalence, up to higher order terms, between the error and a residual type error estimator. Second, we prove that the volumetric part of the residual is dominated by a constant times the edge or face residuals, again up to higher order terms. This result was not known for eigenvalue problems.

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