Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

summary:The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, $Q_{1}^{\rm rot}$, $EQ_{1}^{\rm rot}$ and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results

Guaranteed Lower Eigenvalue Bound of Steklov Operator with Conforming Finite Element Methods

For the eigenvalue problem of the Steklov differential operator, by following Liu's approach, an algorithm utilizing the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed method requires the a priori error estimation for FEM solution to nonhomogeneous Neumann problems, which is solved by constructing the hypercircle for the corresponding FEM spaces and boundary conditions. Numerical examples are also shown to confirm the efficiency of our proposed method.Comment: 21 pages, 4 figures, 4 table

A posteriori Error Estimators based on Duality Techniques from the Calculus of Variations

A theoretical framework is presented within which we can systematically develop a posteriori error estimators for a quite general class of variational statements, involving a linear operator and two convex functionals. We merely require, that the linear operator be coercive and the corresponding functional be uniformly convex. As the second functional may be arbitrary, the theory can also cover constrained variational formulations. Two applications are discussed in detail: the Dirichlet Problem and the Obstacle Problem. A number of technical issues is considered, which pertain to the evaluation of the proposed error bounds using finite element methods: Inter alia a novel non-conforming discretisation scheme for the dual formulation is analysed. The resulting algebraic problem may be solved by a new preconditioned relaxation method, for which a proof of convergence is supplied

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described