Method For Variational Problems Research Articles

A Partition of Unity construction of the stabilization function in Nitsche’s method for variational problems

In this paper we develop a partition-of-unity construction of the stabilization function required in Nitsche’s method, which can be seen as a generalization of the element-wise construction that is widely used in finite element methods. This allows for the use of Nitsche’s method within the Partition of Unity Method with a stabilization function that is not simply a constant over the whole boundary. In addition to that, we introduce a patch-aggregation approach designed to avoid arbitrarily large values of the stabilization function and the associated ill-conditioned systems and deteriorated convergence rates. We present numerical results to validate the proposed methods, covering Dirichlet boundary conditions, interface constraints and higher-order problems. These results clearly show that our approach leads to optimal convergence rates.

Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap

We investigate the convergence of the Crouzeix-Raviart finite element method for variational problems with non-autonomous integrands that exhibit non-standard growth conditions. While conforming schemes fail due to the Lavrentiev gap phenomenon, we prove that the solution of the Crouzeix-Raviart scheme converges to a global minimiser. Numerical experiments illustrate the performance of the scheme and give additional analytical insights.

The elements of parametrical regulation theory for computable general equilibrium models

PurposeThe purpose of this paper is to present some results of the development of parametrical regulation theory elements for computable general equilibrium models (CGE models), taking into consideration their peculiarities.Design/methodology/approachTheoretical results have been obtained by means of applying geometrical methods for variational problems and methods of the theory of discrete dynamic systems. These results have been used for solving concrete practical problems.FindingsThe authors proved a statement on the existence of solution for variational calculus problem on the choice of the optimal laws of parametrical regulation within the given finite set of algorithms for discrete dynamic systems. A statement has been proved on sufficient conditions for the existence of an extremal's bifurcation point of variational calculus problem on the choice of the optimum laws of parametrical regulation within the given finite set of algorithms for discrete dynamic systems. Optimal laws of parametrical regulation (on the level of one and two parameters) of economic system evolution on the basis of the examined mathematical model have been defined. The bifurcation line was constructed for the given area of uncontrolled parameter values.Practical implicationsThe research results can be applied for the choice and realization of an effective state budget and tax policy.Originality/valueThe paper elaborates the elements of parametrical regulation theory of economic system development on the basis of CGE models and shows the effectiveness of parametrical regulation theory application on the example of one CGE model.

Bregman-like functions and proximal methods for variational problems with nonlinear constraints†

The use of Bregman functions, entropic ϕ-divergence or logarithmic-quadratic kernels, allows to construct a series of generalized proximal methods for the stable solution of convex optimization problems and variational inequalities with maximal monotone operators. The key advantage of these methods in comparison to the classical proximal regularization is that the auxiliary problems are structurally simpler than the original ones, in particular, they result in unconstrained inclusions. But, such methods with “interior point effect” were developed so far only for linearly constrained problems. In the present article, the use of Bregman functions with a modified “convergence sensing condition” enables us to construct an interior proximal method for solving variational inequalities (with multi-valued operators) on nonpolyhedral sets. The convergence results admit a successive approximation of the multi-valued operator (by means of the concept of ε-enlargements) and an inexact calculation of proximal iterates. †Dedicated to the 65th birthday of Diethard Pallaschke.

Direct Walsh-wavelet packet method for variational problems

A direct Ritz method for solving variational problems is demonstrated. Walsh-wavelet packet bases are used as the basis functions. It is shown how the form of the operational matrix of integration affects the accuracy of the obtained approximate solutions. One or two approximation fields are formulated and the method of Lagrange multipliers is used to solve illustrative problems with free and constrained ends.

Construction and Convergence Analysis of Iterative Methods for Variational Problems with a Nondifferentiable Functional

Construction and Convergence Analysis of Iterative Methods for Variational Problems with a Nondifferentiable Functional

An algebraic study of a local multigrid method for variational problems

A two-level iterative method for solving systems of equations arising from local refinement of finite element meshes is proposed. The smoothing steps consists of Gauss-Seidel relaxations applied only to the local mesh. A direct solver is required on the global coarse mesh. We analyze this scheme by modifying the standard algebraic convergence theory. The estimates of the convergence factor do not require any regularity assumption. The estimates can be evaluated element-by-element. The evaluated estimates are compared with the actual convergence factor and with another bound in several numerical examples. The theory also applies to a method with a direct solver on the local fine mesh. In this case we have obtained very sharp (and in fact optimal) estimates of the convergence factors.

Shifted-Jacobi series direct method for variational problems

The operational matrix or integration of a Jacobi vector whose elements are Jacobi polynomials is introduced and applied to solve variational problems by a direct method. Illustrative examples are given to demonstrate that only a small number of shifted-Jacobi polynomials are needed to obtain an accurate solution.

The Variational Method for Problems of Radiative Transfer. II. The Formation of Absorption Lines in the Milne-Eddington Model.

The Variational Method for Problems of Radiative Transfer. II. The Formation of Absorption Lines in the Milne-Eddington Model.

The Variational Method for Problems of Neutron Diffusion and of Radiative Transfer

The functional as proposed by R. E. Marshak for solving certain inhomogeneous integral equations by the variational method is modified in such a way that the simultaneous equations which determine the parameters become linear. Thus we can solve this type of integral equations with an accuracy as high as required without much labor. As an illustration we obtain by the present method an approximate solution of Milne's integral equation for the neutron density (or for the source function in the problem of radiative transfer) in a simple form. Our approximate solution agrees (for the first four figures) everywhere with the one computed by C. Mark from the exact solution.

On a Generalization of the Variational Method for Problems of Radiative Transfer.

On a Generalization of the Variational Method for Problems of Radiative Transfer.