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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.

On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations

Xuejun Xu, C. O. Chow, S. H. Lui (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, a Dirichlet-Neumann substructuring domain decomposition method is presented for a finite element approximation to the nonlinear Navier-Stokes equations. It is shown that the Dirichlet-Neumann domain decomposition sequence converges geometrically to the true solution provided the Reynolds number is sufficiently small. In this method, subdomain problems are linear. Other version where the subdomain problems are linear Stokes problems is also presented.

On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations

Xuejun Xu, C. O. Chow, S. H. Lui (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, a Dirichlet-Neumann substructuring domain decomposition method is presented for a finite element approximation to the nonlinear Navier-Stokes equations. It is shown that the Dirichlet-Neumann domain decomposition sequence converges geometrically to the true solution provided the Reynolds number is sufficiently small. In this method, subdomain problems are linear. Other version where the subdomain problems are linear Stokes problems is also presented.

On numerical solution of weight minimization of elastic bodies weakly supporting tension

Petr Kočandrle, Petr Rybníček (1995)

Applications of Mathematics

Shape optimization of a two-dimensional elastic body is considered, provided the material is weakly supporting tension. The problem generalizes that of a masonry dam subjected to its weight and to the hydrostatic pressure. A part of the boundary has to be found so as to minimize a given cost functional. The numerical realization using a penalty method and finite element technique is presented. Some typical results are shown.

On phase segregation in nonlocal two-particle Hartree systems

Walter Aschbacher, Marco Squassina (2009)

Open Mathematics

We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.

On polynomial robustness of flux reconstructions

Miloslav Vlasák (2020)

Applications of Mathematics

We deal with the numerical solution of elliptic not necessarily self-adjoint problems. We derive a posteriori upper bound based on the flux reconstruction that can be directly and cheaply evaluated from the original fluxes and we show for one-dimensional problems that local efficiency of the resulting a posteriori error estimators depends on p 1 / 2 only, where p is the discretization polynomial degree. The theoretical results are verified by numerical experiments.

On semiregular families of triangulations and linear interpolation

Michal Křížek (1991)

Applications of Mathematics

We consider triangulations formed by triangular elements. For the standard linear interpolation operator π h we prove the interpolation order to be v - π h v 1 , p ≤ C h v 2 , p for p > 1 provided the corresponding family of triangulations is only semiregular. In such a case the well-known Zlámal’s condition upon the minimum angle need not be satisfied.

On solution to an optimal shape design problem in 3-dimensional linear magnetostatics

Dalibor Lukáš (2004)

Applications of Mathematics

In this paper we present theoretical, computational, and practical aspects concerning 3-dimensional shape optimization governed by linear magnetostatics. The state solution is approximated by the finite element method using Nédélec elements on tetrahedra. Concerning optimization, the shape controls the interface between the air and the ferromagnetic parts while the whole domain is fixed. We prove the existence of an optimal shape. Then we state a finite element approximation to the optimization...

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