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Guaranteed a-posteriori error estimation for finite element solutions of nonstationary heat conduction problems based on their elliptic reconstructions

Theofanis Strouboulis, Delin Wang (2024)

Applications of Mathematics

We deal with the a-posteriori estimation of the error for finite element solutions of nonstationary heat conduction problems with mixed boundary conditions on bounded polygonal domains. The a-posteriori error estimates are constucted by solving stationary “reconstruction” problems, obtained by replacing the time derivative of the exact solution by the time derivative of the finite element solution. The main result is that the reconstructed solutions, or reconstructions, are superconvergent approximations...

Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method

Martin Ladecký, Ivana Pultarová, Jan Zeman (2021)

Applications of Mathematics

A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite...

h p -FEM for three-dimensional elastic plates

Monique Dauge, Christoph Schwab (2002)

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

In this work, we analyze hierarchic h p -finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the h p -FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate. We prove that, as the plate half-thickness ε tends to zero, the h p -discretization is consistent with the three-dimensional solution to any power of ε in the energy...

H P -finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form

Andrea Toselli (2003)

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

We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect...

Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

Owe Axelsson, János Karátson (2013)

Open Mathematics

A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

Hermite pseudospectral method for nonlinear partial differential equations

Ben-yu Guo, Cheng-long Xu (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Hermite polynomial interpolation is investigated. Some approximation results are obtained. As an example, the Burgers equation on the whole line is considered. The stability and the convergence of proposed Hermite pseudospectral scheme are proved strictly. Numerical results are presented.

Hexahedral H(div) and H(curl) finite elements

Richard S. Falk, Paolo Gatto, Peter Monk (2011)

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

We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using...

Hexahedral H(div) and H(curl) finite elements*

Richard S. Falk, Paolo Gatto, Peter Monk (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron...

High order edge elements on simplicial meshes

Francesca Rapetti (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

Low order edge elements are widely used for electromagnetic field problems. Higher order edge approximations are receiving increasing interest but their definition become rather complex. In this paper we propose a simple definition for Whitney edge elements of polynomial degree higher than one. We give a geometrical localization of all degrees of freedom over particular edges and provide a basis for these elements on simplicial meshes. As for Whitney edge elements of degree one, the basis is...

High order transmission conditions for thin conductive sheets in magneto-quasistatics

Kersten Schmidt, Sébastien Tordeux (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...

High order transmission conditions for thin conductive sheets in magneto-quasistatics

Kersten Schmidt, Sébastien Tordeux (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...

Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type

Liping Liu, Michal Křížek, Pekka Neittaanmäki (1996)

Applications of Mathematics

A nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions is examined. The problem describes for instance a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree k ≥ 1 we prove the optimal rates of convergence 𝒪 ( h k ) in the H 1 -norm and 𝒪 ( h k + 1 ) in the L 2 -norm provided the true solution is sufficiently smooth. Considerations are restricted to domains with polyhedral boundaries. Numerical integration is not taken into account....

Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time

Alexei Lozinski, Jacek Narski, Claudia Negulescu (2014)

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

This paper deals with the numerical study of a nonlinear, strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is second-order accurate in both, temporal and spacial variables. The discretization in time is done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to be independent of the anisotropy parameter , and this for fixed...

How to recover the gradient of linear elements on nonuniform triangulations

Ivan Hlaváček, Michal Křížek, Vladislav Pištora (1996)

Applications of Mathematics

We propose and examine a simple averaging formula for the gradient of linear finite elements in R d whose interpolation order in the L q -norm is 𝒪 ( h 2 ) for d < 2 q and nonuniform triangulations. For elliptic problems in R 2 we derive an interior superconvergence for the averaged gradient over quasiuniform triangulations. A numerical example is presented.

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