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On simplicial red refinement in three and higher dimensions

Korotov, SergeyKřížek, Michal — 2013

Applications of Mathematics 2013

We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original...

Two-sided a posteriori estimates of global and local errors for linear elliptic type boundary value problems

Hannukainen, AnttiKorotov, Sergey — 2006

Programs and Algorithms of Numerical Mathematics

The paper is devoted to the problem of reliable control of accuracy of approximate solutions obtained in computer simulations. This task is strongly related to the so-called a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain, where such errors are too large and certain mesh refinements should be performed. Mathematical model described by a linear elliptic (reaction-diffusion) equation with mixed boundary conditions is considered....

Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions

Sergey Korotov — 2007

Applications of Mathematics

The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered...

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

János KarátsonSergey Korotov — 2009

Applications of Mathematics

The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed...

Generalization of the Zlámal condition for simplicial finite elements in d

Jan BrandtsSergey KorotovMichal Křížek — 2011

Applications of Mathematics

The famous Zlámal’s minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in 2 d . In this paper we present and discuss its generalization to simplicial partitions in any space dimension.

Simplicial finite elements in higher dimensions

Jan BrandtsSergey KorotovMichal Křížek — 2007

Applications of Mathematics

Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the...

Nonobtuse tetrahedral partitions that refine locally towards Fichera-like corners

Larisa BeilinaSergey KorotovMichal Křížek — 2005

Applications of Mathematics

Linear tetrahedral finite elements whose dihedral angles are all nonobtuse guarantee the validity of the discrete maximum principle for a wide class of second order elliptic and parabolic problems. In this paper we present an algorithm which generates nonobtuse face-to-face tetrahedral partitions that refine locally towards a given Fichera-like corner of a particular polyhedral domain.

Galerkin approximations for the linear parabolic equation with the third boundary condition

István FaragóSergey KorotovPekka Neittaanmäki — 2003

Applications of Mathematics

We solve a linear parabolic equation in d , d 1 , with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the θ -method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.

On Synge-type angle condition for d -simplices

Antti HannukainenSergey KorotovMichal Křížek — 2017

Applications of Mathematics

The maximum angle condition of J. L. Synge was originally introduced in interpolation theory and further used in finite element analysis and applications for triangular and later also for tetrahedral finite element meshes. In this paper we present some of its generalizations to higher-dimensional simplicial elements. In particular, we prove optimal interpolation properties of linear simplicial elements in d that degenerate in some way.

On interpolation error on degenerating prismatic elements

Ali KhademiSergey KorotovJon 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.

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