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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...

Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation

Weihua Geng, Shan Zhao (2013)

Molecular Based Mathematical Biology

The Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator splitting...

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...

Generalized combined field integral equations for the iterative solution of the three-dimensional Helmholtz equation

Xavier Antoine, Marion Darbas (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper addresses the derivation of new second-kind Fredholm combined field integral equations for the Krylov iterative solution of tridimensional acoustic scattering problems by a smooth closed surface. These integral equations need the introduction of suitable tangential square-root operators to regularize the formulations. Existence and uniqueness occur for these formulations. They can be interpreted as generalizations of the well-known Brakhage-Werner [A. Brakhage and P. Werner, Arch....

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...

Gradient-free and gradient-based methods for shape optimization of water turbine blade

Bastl, Bohumír, Brandner, Marek, Egermaier, Jiří, Horníková, Hana, Michálková, Kristýna, Turnerová, Eva (2019)

Programs and Algorithms of Numerical Mathematics

The purpose of our work is to develop an automatic shape optimization tool for runner wheel blades in reaction water turbines, especially in Kaplan turbines. The fluid flow is simulated using an in-house incompressible turbulent flow solver based on recently introduced isogeometric analysis (see e.g. J. A. Cotrell et al.: Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, 2009). The proposed automatic shape optimization approach is based on a so-called hybrid optimization which combines...

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...

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