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On a variant of the local projection method stable in the SUPG norm

Petr Knobloch (2009)

Kybernetika

We consider the local projection finite element method for the discretization of a scalar convection-diffusion equation with a divergence-free convection field. We introduce a new fluctuation operator which is defined using an orthogonal L 2 projection with respect to a weighted L 2 inner product. We prove that the bilinear form corresponding to the discrete problem satisfies an inf-sup condition with respect to the SUPG norm and derive an error estimate for the discrete solution.

On approximation of the Neumann problem by the penalty method

Michal Křížek (1993)

Applications of Mathematics

We prove that penalization of constraints occuring in the linear elliptic Neumann problem yields directly the exact solution for an arbitrary set of penalty parameters. In this case there is a continuum of Lagrange's multipliers. The proposed penalty method is applied to calculate the magnetic field in the window of a transformer.

On discontinuous Galerkin method and semiregular family of triangulations

Aleš Prachař (2006)

Applications of Mathematics

Discretization of second order elliptic partial differential equations by discontinuous Galerkin method often results in numerical schemes with penalties. In this paper we analyze these penalized schemes in the context of quite general triangular meshes satisfying only a semiregularity assumption. A new (modified) penalty term is presented and theoretical properties are proven together with illustrative numerical results.

On equilibrium finite elements in three-dimensional case

Sergey Korotov (1997)

Applications of Mathematics

The space of divergence-free functions with vanishing normal flux on the boundary is approximated by subspaces of finite elements that have the same property. The easiest way of generating basis functions in these subspaces is considered.

On exact results in the finite element method

Ivan Hlaváček, Michal Křížek (2001)

Applications of Mathematics

We prove that the finite element method for one-dimensional problems yields no discretization error at nodal points provided the shape functions are appropriately chosen. Then we consider a biharmonic problem with mixed boundary conditions and the weak solution u . We show that the Galerkin approximation of u based on the so-called biharmonic finite elements is independent of the values of u in the interior of any subelement.

On FE-grid relocation in solving unilateral boundary value problems by FEM

Jaroslav Haslinger, Pekka Neittaanmäki, Kimmo Salmenjoki (1992)

Applications of Mathematics

We consider FE-grid optimization in elliptic unilateral boundary value problems. The criterion used in grid optimization is the total potential energy of the system. It is shown that minimization of this cost functional means a decrease of the discretization error or a better approximation of the unilateral boundary conditions. Design sensitivity analysis is given with respect to the movement of nodal points. Numerical results for the Dirichlet-Signorini problem for the Laplace equation and the...

On Finite Element Methods for 2nd order (semi–) periodic Eigenvalue Problems

De Schepper, H. (2000)

Serdica Mathematical Journal

We deal with a class of elliptic eigenvalue problems (EVPs) on a rectangle Ω ⊂ R^2 , with periodic or semi–periodic boundary conditions (BCs) on ∂Ω. First, for both types of EVPs, we pass to a proper variational formulation which is shown to fit into the general framework of abstract EVPs for symmetric, bounded, strongly coercive bilinear forms in Hilbert spaces, see, e.g., [13, §6.2]. Next, we consider finite element methods (FEMs) without and with numerical quadrature. The aim of the paper is...

On interpolation error on degenerating prismatic elements

Ali Khademi, Sergey Korotov, Jon Eivind Vatne (2018)

Applications of Mathematics

We propose an analogue of the maximum angle condition (commonly used in finite element analysis for triangular and tetrahedral meshes) for the case of prismatic elements. Under this condition, prisms in the meshes may degenerate in certain ways, violating the so-called inscribed ball condition presented by P. G. Ciarlet (1978), but the interpolation error remains of the order O ( h ) in the H 1 -norm for sufficiently smooth functions.

On iterative solution of nonlinear heat-conduction and diffusion problems

Herbert Gajewski (1977)

Aplikace matematiky

The present paper deals with the numerical solution of the nonlinear heat equation. An iterative method is suggested in which the iterations are obtained by solving linear heat equation. The convergence of the method is proved under very natural conditions on given input data of the original problem. Further, questions of convergence of the Galerkin method applied to the original equation as well as to the linear equations in the above mentioned iterative method are studied.

Currently displaying 961 – 980 of 1415