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Finite element methods on non-conforming grids by penalizing the matching constraint

Eric Boillat (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The present paper deals with a finite element approximation of partial differential equations when the domain is decomposed into sub-domains which are meshed independently. The method we obtain is never conforming because the continuity constraints on the boundary of the sub-domains are not imposed strongly but only penalized. We derive a selection rule for the penalty parameter which ensures a quasi-optimal convergence.

Finite element modeling of wood structure

Koňas, Petr (2008)

Programs and Algorithms of Numerical Mathematics

This work is focused on a weak solution of a coupled physical task of the microwave wood drying process with stress-strain effects and moisture/temperature dependency. Due to the well known weak solutions for the individual physical fields, the author concerns with the coupled stress-strain relation coupled with the moisture and temperature distributions. For the scale dependency the subgrid upscaling method was used. The solved region is assumed to be divided into discontinuous subregions according...

Finite element modelling of flow and temperature regime in shallow lakes

Podsechin, Victor, Schernewski, Gerald (2013)

Applications of Mathematics 2013

A two-dimensional depth-averaged flow and temperature model was applied to study the circulation patterns in the Oder (Szczecin) Lagoon located on the border between Germany and Poland. The system of shallow water and temperature evolution equations is discretized with the modified Utnes scheme [4], which is characterized by a semi-decoupling algorithm. The continuity equation is rearranged to Helmholtz equation form. The upwinding Tabata method [3] is used to approximate convective terms. Averaged...

Finite element modelling of some incompressible fluid flow problems

Burda, Pavel, Novotný, Jaroslav, Šístek, Jakub, Damašek, Alexandr (2008)

Programs and Algorithms of Numerical Mathematics

We deal with modelling of flows in channels or tubes with abrupt changes of the diameter. The goal of this work is to construct the FEM solution in the vicinity of these corners as precise as desired. We present two ways. The first approach makes use of a posteriori error estimates and the adaptive strategy. The second approach is based on the asymptotic behaviour of the exact solution in the vicinity of the corner and on the a priori error estimate of the FEM solution. Then we obtain the solution...

Finite element solution of a hyperbolic-parabolic problem

Rudolf Hlavička (1994)

Applications of Mathematics

Existence and finite element approximation of a hyperbolic-parabolic problem is studied. The original two-dimensional domain is approximated by a polygonal one (external approximations). The time discretization is obtained using Euler’s backward formula (Rothe’s method). Under certain smoothing assumptions on the data (see (2.6), (2.7)) the existence and uniqueness of the solution and the convergence of Rothe’s functions in the space C ( I ¯ , V ) is proved.

Finite element solution of a stationary heat conduction equation with the radiation boundary condition

Zdeněk Milka (1993)

Applications of Mathematics

In this paper we present a weak formulation of a two-dimensional stationary heat conduction problem with the radiation boundary condition. The problem can be described by an operator which is monotone on the convex set of admissible functions. The relation between classical and weak solutions as well as the convergence of the finite element method to the weak solution in the norm of the Sobolev space H 1 ( Ω ) are examined.

Finite element solution of flows through cascades of profiles in a layer of variable thickness

Miloslav Feistauer, Jiří Felcman, Zdeněk Vlášek (1986)

Aplikace matematiky

The paper is devoted to the numerical modelling of a subsonic irrotational nonviscous flow past a cascade of profiles in a variable thickness fluid layer. It leads to a nonlinear two-dimensional elliptic problem with nonstandard nonhomogeneous boundary conditions. The problem is discretized by the finite element method. Both theoretical and practical questions of the finite element implementation are studied; convergence of the method, numerical integration, iterative methods for the solution of...

Finite element solution of the fundamental equations of semiconductor devices. II

Miloš Zlámal (2001)

Applications of Mathematics

In part I of the paper (see Zlámal [13]) finite element solutions of the nonstationary semiconductor equations were constructed. Two fully discrete schemes were proposed. One was nonlinear, the other partly linear. In this part of the paper we justify the nonlinear scheme. We consider the case of basic boundary conditions and of constant mobilities and prove that the scheme is unconditionally stable. Further, we show that the approximate solution, extended to the whole time interval as a piecewise...

Finite element variational crimes in the case of semiregular elements

Alexander Ženíšek (1996)

Applications of Mathematics

The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain Ω whose boundary ∂ Ω is formed by two circles Γ 1 , Γ 2 with the same center S 0 and radii R 1 , R 2 = R 1 + ϱ , where ϱ ≪ R 1 . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for u = 0 are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus triangles obeying...

Finite element-based observer design for nonlinear systems with delayed measurements

Branislav Rehák (2019)

Kybernetika

This paper presents a computational procedure for the design of an observer of a nonlinear system. Outputs can be delayed, however, this delay must be known and constant. The characteristic feature of the design procedure is computation of a solution of a partial differential equation. This equation is solved using the finite element method. Conditions under which existence of a solution is guaranteed are derived. These are formulated by means of theory of partial differential equations in L 2 -space....

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