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Variational problems in domains with cusp points

Alexander Ženíšek (1993)

Applications of Mathematics

The finite element analysis of linear elliptic problems in two-dimensional domains with cusp points (turning points) is presented. This analysis needs on one side a generalization of results concerning the existence and uniqueness of the solution of a constinuous elliptic variational problem in a domain the boundary of which is Lipschitz continuous and on the other side a presentation of a new finite element interpolation theorem and other new devices.

Variational sensitivity analysis of parametric Markovian market models

Norbert Hilber, Christoph Schwab, Christoph Winter (2008)

Banach Center Publications

Parameter sensitivities of prices for derivative contracts play an important role in model calibration as well as in quantification of model risk. In this paper a unified approach to the efficient numerical computation of all sensitivities for Markovian market models is presented. Variational approximations of the integro-differential equations corresponding to the infinitesimal generators of the market model differentiated with respect to the model parameters are employed. Superconvergent approximations...

Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method

A. Klöckner, T. Warburton, J. S. Hesthaven (2011)

Mathematical Modelling of Natural Phenomena

We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechanism. Using an artificial viscosity in the latter role, we obtain a DG scheme for the numerical solution of nonlinear systems of conservation laws. Building on work by Persson and Peraire, we thoroughly justify the...

Wavelet bases for the biharmonic problem

Bímová, Daniela, Černá, Dana, Finěk, Václav (2013)

Programs and Algorithms of Numerical Mathematics

In our contribution, we study different Riesz wavelet bases in Sobolev spaces based on cubic splines satisfying homogeneous Dirichlet boundary conditions of the second order. These bases are consequently applied to the numerical solution of the biharmonic problem and their quantitative properties are compared.

Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

Nils Reich (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

For a class of anisotropic integrodifferential operators ℬ arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations ℬ u = f on [0,1]n with possibly large n. Under certain conditions on ℬ , the scheme is of essentially optimal and dimension independent complexity 𝒪 (h-1| log h |2(n-1)) without corrupting the convergence or smoothness requirements...

Weaker convergence conditions for the secant method

Ioannis K. Argyros, Saïd Hilout (2014)

Applications of Mathematics

We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.

Weight minimization of an elastic plate with a unilateral inner obstacle by a mixed finite element method

Ivan Hlaváček (1994)

Applications of Mathematics

Unilateral deflection problem of a clamped plate above a rigid inner obstacle is considered. The variable thickness of the plate is to be optimized to reach minimal weight under some constraints for maximal stresses. Since the constraints are expressed in terms of the bending moments only, Herrmann-Hellan finite element scheme is employed. The existence of an optimal thickness is proved and some convergence analysis for approximate penalized optimal design problem is presented.

Weight minimization of elastic bodies weakly supporting tension. II. Domains with two curved sides

Ivan Hlaváček, Michal Křížek (1992)

Applications of Mathematics

Extending the results of the previous paper [1], the authors consider elastic bodies with two design variables, i.e. "curved trapezoids" with two curved variable sides. The left side is loaded by a hydrostatic pressure. Approximations of the boundary are defined by cubic Hermite splines and piecewise linear finite elements are used for the displacements. Both existence and some convergence analysis is presented for approximate penalized optimal design problems.

Weight minimization of elastic bodies weakly supporting tension. I. Domains with one curved side

Ivan Hlaváček, Michal Křížek (1992)

Applications of Mathematics

Shape optimization of a two-dimensional elastic body is considered, provided the material is weakly supporting tension. The problem generalizes that of a masonry dam subjected to its own weight and to the hydrostatic presure. Existence of an optimal shape is proved. Using a penalty method and finite element technique, approximate solutions are proposed and their convergence is analyzed.

Weight minimization of elastic plates using Reissner-Mindlin model and mixed-interpolated elements

Ivan Hlaváček (1996)

Applications of Mathematics

The problem to find an optimal thickness of the plate in a set of bounded Lipschitz continuous functions is considered. Mean values of the intensity of shear stresses must not exceed a given value. Using a penalty method and finite element spaces with interpolation to overcome the “locking” effect, an approximate optimization problem is proposed. We prove its solvability and present some convergence analysis.

Weighted regularization for composite materials in electromagnetism

Patrick Ciarlet Jr., François Lefèvre, Stéphanie Lohrengel, Serge Nicaise (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem...

What is the smallest possible constant in Céa's lemma?

Wei Chen, Michal Křížek (2006)

Applications of Mathematics

We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in ℝ d with d ∈ { 1 , 2 , 3 , ... } . The constant C ≥ 1 appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and...

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