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From h to p Efficiently: Selecting the Optimal Spectral/hp Discretisation in Three Dimensions

C. D. Cantwell, S. J. Sherwin, R. M. Kirby, P. H. J. Kelly (2011)

Mathematical Modelling of Natural Phenomena

There is a growing interest in high-order finite and spectral/hp element methods using continuous and discontinuous Galerkin formulations. In this paper we investigate the effect of h- and p-type refinement on the relationship between runtime performance and solution accuracy. The broad spectrum of possible domain discretisations makes establishing a performance-optimal selection non-trivial. Through comparing the runtime of different implementations...

Functional a posteriori error estimates for incremental models in elasto-plasticity

Sergey Repin, Jan Valdman (2009)

Open Mathematics

We consider incremental problem arising in elasto-plastic models with isotropic hardening. Our goal is to derive computable and guaranteed bounds of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. Such estimates are obtained by an advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals [24, 30]. They do no contain mesh-dependent constants...

Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems

Martin Kahlbacher, Stefan Volkwein (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of linear and non-linear systems. It is based on a Galerkin type discretization with basis elements created from the system itself. In this work, error estimates for Galerkin POD methods for linear elliptic, parameter-dependent systems are proved. The resulting error bounds depend on the number of POD basis functions and on the parameter grid that is used to generate the snapshots and to compute the POD basis. The...

Generalization of the Zlámal condition for simplicial finite elements in ℝ d

Jan Brandts, Sergey Korotov, Michal 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.

Generalizations of the Finite Element Method

Marc Schweitzer (2012)

Open Mathematics

This paper is concerned with the generalization of the finite element method via the use of non-polynomial enrichment functions. Several methods employ this general approach, e.g. the extended finite element method and the generalized finite element method. We review these approaches and interpret them in the more general framework of the partition of unity method. Here we focus on fundamental construction principles, approximation properties and stability of the respective numerical method. To...

Generic implementation of finite element methods in the Distributed and Unified Numerics Environment (DUNE)

Peter Bastian, Felix Heimann, Sven Marnach (2010)

Kybernetika

In this paper we describe PDELab, an extensible C++ template library for finite element methods based on the Distributed and Unified Numerics Environment (Dune). PDELab considerably simplifies the implementation of discretization schemes for systems of partial differential equations by setting up global functions and operators from a simple element-local description. A general concept for incorporation of constraints eases the implementation of essential boundary conditions, hanging nodes and varying...

Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations

M. R. Swager, Y. C. Zhou (2013)

Molecular Based Mathematical Biology

A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-to-one correspondence between the continuous piecewise polynomial space of degree k + 1 and the divergencefree vector space of degree k, one...

Goal-oriented error estimates including algebraic errors in discontinuous Galerkin discretizations of linear boundary value problems

Vít Dolejší, Filip Roskovec (2017)

Applications of Mathematics

We deal with a posteriori error control of discontinuous Galerkin approximations for linear boundary value problems. The computational error is estimated in the framework of the Dual Weighted Residual method (DWR) for goal-oriented error estimation which requires to solve an additional (adjoint) problem. We focus on the control of the algebraic errors arising from iterative solutions of algebraic systems corresponding to both the primal and adjoint problems. Moreover, we present two different reconstruction...

Gravimetric quasigeoid in Slovakia by the finite element method

Zuzana Fašková, Karol Mikula, Róbert Čunderlík, Juraj Janák, Michal Šprlák (2007)

Kybernetika

The paper presents the solution to the geodetic boundary value problem by the finite element method in area of Slovak Republic. Generally, we have made two numerical experiments. In the first one, Neumann BC in the form of gravity disturbances generated from EGM-96 is used and the solution is verified by the quasigeoidal heights generated directly from EGM-96. In the second one, Neumann BC is computed from gravity measurements and the solution is compared to the quasigeoidal heights obtained by...

Guaranteed and fully computable two-sided bounds of Friedrichs’ constant

Vejchodský, Tomáš (2013)

Programs and Algorithms of Numerical Mathematics

This contribution presents a general numerical method for computing lower and upper bound of the optimal constant in Friedrichs’ inequality. The standard Rayleigh-Ritz method is used for the lower bound and the method of 𝑎 𝑝𝑟𝑖𝑜𝑟𝑖 - 𝑎 𝑝𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟𝑖 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑖𝑒𝑠 is employed for the upper bound. Several numerical experiments show applicability and accuracy of this approach.

Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems

Ibrahim Cheddadi, Radek Fučík, Mariana I. Prieto, Martin Vohralík (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We derive a posteriori error estimates for singularly perturbed reaction–diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature...

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...

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