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Strongly regular family of boundary-fitted tetrahedral meshes of bounded C 2 domains

Radim Hošek (2016)

Applications of Mathematics

We give a constructive proof that for any bounded domain of the class C 2 there exists a strongly regular family of boundary-fitted tetrahedral meshes. We adopt a refinement technique introduced by Křížek and modify it so that a refined mesh is again boundary-fitted. An alternative regularity criterion based on similarity with the Sommerville tetrahedron is used and shown to be equivalent to other standard criteria. The sequence of regularities during the refinement process is estimated from below...

Superapproximation of the partial derivatives in the space of linear triangular and bilinear quadrilateral finite elements

Dalík, Josef (2013)

Programs and Algorithms of Numerical Mathematics

A method for the second-order approximation of the values of partial derivatives of an arbitrary smooth function u = u ( x 1 , x 2 ) in the vertices of a conformal and nonobtuse regular triangulation 𝒯 h consisting of triangles and convex quadrilaterals is described and its accuracy is illustrated numerically. The method assumes that the interpolant Π h ( u ) in the finite element space of the linear triangular and bilinear quadrilateral finite elements from 𝒯 h is known only.

Superconvergence analysis and a posteriori error estimation of a Finite Element Method for an optimal control problem governed by integral equations

Ningning Yan (2009)

Applications of Mathematics

In this paper, we discuss the numerical simulation for a class of constrained optimal control problems governed by integral equations. The Galerkin method is used for the approximation of the problem. A priori error estimates and a superconvergence analysis for the approximation scheme are presented. Based on the results of the superconvergence analysis, a recovery type a posteriori error estimator is provided, which can be used for adaptive mesh refinement.

Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

Xu Yin, Waixiang Cao, Zhimin Zhang (2024)

Applications of Mathematics

We present a unified approach to studying the superconvergence property of the spectral volume (SV) method for high-order time-dependent partial differential equations using the local discontinuous Galerkin formulation. We choose the diffusion and third-order wave equations as our models to illustrate approach and the main idea. The SV scheme is designed with control volumes constructed using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes...

Superconvergence by Steklov averaging in the finite element method

Karel Kolman (2005)

Applicationes Mathematicae

The Steklov postprocessing operator for the linear finite element method is studied. Superconvergence of order 𝓞(h²) is proved for a class of second order differential equations with zero Dirichlet boundary conditions for arbitrary space dimensions. Relations to other postprocessing and averaging schemes are discussed.

Superconvergence of a stabilized approximation for the Stokes eigenvalue problem by projection method

Pengzhan Huang (2014)

Applications of Mathematics

This paper presents a superconvergence result based on projection method for stabilized finite element approximation of the Stokes eigenvalue problem. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The paper complements the work of Li et al. (2012), which establishes the superconvergence result of the Stokes equations by the stabilized finite element method. Moreover, numerical tests confirm the theoretical analysis.

Superconvergence of mixed finite element semi-discretizations of two time-dependent problems

Jan Brandts (1999)

Applications of Mathematics

We will show that some of the superconvergence properties for the mixed finite element method for elliptic problems are preserved in the mixed semi-discretizations for a diffusion equation and for a Maxwell equation in two space dimensions. With the help of mixed elliptic projection we will present estimates global and pointwise in time. The results for the Maxwell equations form an extension of existing results. For both problems, our results imply that post-processing and a posteriori error estimation...

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