Auxiliary Linear Problem Research Articles

Superconvergence of rectangular finite element with interpolated coefficients for semilinear elliptic problem

In this paper, the rectangular finite element method with interpolated coefficients for the second-order semilinear elliptic problem is introduced and analyzed. Assume that Ω is polygonal domain, and rectangular partition is quasiuniform and denote by Z 0 and Z 1 the sets of ( n + 1)-order Lobatto points and n-order Gauss points of all elements, respectively. Based on an orthogonal expansion in an element, and on the property of corresponding finite element for an auxiliary linear elliptic problem, optimal superconvergence u − u h = O( h n+2 ), n ⩾ 2, at z ∈ Z 0 and D( u − u h ) = O( h n+1 ), n ⩾ 1, at z ∈ Z 1 are proved, respectively. It is shown that the finite element with interpolated coefficients has the same superconvergence as that of classical finite elements, but more economic and efficient. Finally the results in the case of quadratic finite element are supported by a numerical example.

Steady-state creep analysis of pressurized pipe weldments by perturbation method

The stress analysis of pressurized circumferential pipe weldments under steady state creep is considered. The creep response of the material is governed by Norton’s law. Numerical and analytical solutions are obtained by means of perturbation method, the unperturbed solution corresponds to the stress field in a homogeneous pipe. The correction terms are treated as stresses defined with the help of an auxiliary linear elastic problem. Exact expressions for jumps of hoop and radial stresses at the interface are obtained. The proposed technique essentially simplifies parametric analysis of multi-material components.

A HIERARCHICAL A POSTERIORI ERROR ESTIMATE FOR AN ADVECTION-DIFFUSION-REACTION PROBLEM

In this work we introduce a new a posteriori error estimate of hierarchical type for the advection-diffusion-reaction equation. We prove the equivalence between the energy norm of the error and our error estimate using an auxiliary linear problem for the residual and an easy way to prove inf–sup condition.

HOMOGENIZATION OF SEMILINEAR PARABOLIC EQUATIONS IN PERFORATED DOMAINS

This paper is devoted to the homogenization of a semilinear parabolic equation with rapidly oscillating coefficients in a domain periodically perforated by e-periodic holes of size e. A Neumann condition is prescribed on the boundary of the holes. The presence of the holes does not allow to prove a compactness of the solutions in L2. To overcome this difficulty, the authors introduce a suitable auxiliary linear problem to which a corrector result is applied. Then, the asymptotic behaviour of the semilinear problem as e→0 is described, and the limit equation is given.

Functional Equations and Quantum Separation of Variables for 3d Spin Models

The recently proposed invariant formulation of the auxiliary linear problem for 3d integrable models provides several new ideas for solving the spectral problem of 3d spin models, e.g., the Zamolodchikov–Bazhanov–Baxter model in its vertex formulation. This paper announces results following from the invariant formulation. We formulate the class of 3d spin models that are essentially appropriately parameterized inhomogeneous Zamolodchikov–Bazhanov–Baxter models, present an expression for the generating function of the complete set of matrices commuting with the transfer matrix of this model (integrals of motion), give the functional equations defining the eigenvalues of the integrals of motion and the transfer matrices, explicitly describe the groupoid of isospectral transformations of the initial system of integrals of motion, and finally give an explicit parameterization of a projection operator onto the separated states in the sense of the quantum separation of variables (functional Bethe ansatz).

A Proportioning Based Algorithm with Rate of Convergence for Bound Constrained Quadratic Programming

The proportioning algorithm with projections turned out to be an efficient algorithm for iterative solution of large quadratic programming problems with simple bounds and box constraints. Important features of this active set based algorithm are the adaptive precision control in the solution of auxiliary linear problems and capability to add or remove many indices from the active set in one step. In this paper a modification of the algorithm is presented that enables to find its rate of convergence in terms of the spectral condition number of the Hessian matrix and avoid any backtracking. The modified algorithm is shown to preserve the finite termination property of the original algorithm for problems that are not dual degenerate.

Quantum Lax Operators and Discrete 2+1-Dimensional Integrable Models

A specific quantum Lax operator, forming a quantum U q (sl n) chain, is suggested. The intertwining R L L relation is proved. The invariant local 2+1-dimensional reformulation of the auxiliary linear problem for the L-operator is discussed.

Resonance and multiplicity in periodic boundary value problems with singularity

The paper deals with the boundary value problem \[ u^{\prime \prime }+k\,u=g(u)+e(t),\quad u(0)=u(2\pi ),\,\,u^{\prime }(0)=u^{\prime }(2\pi ), \] where $k\in \mathbb{R}$, $g\:I\mapsto \mathbb{R}$ is continuous, $e\in \mathbb{L}J$ and $\lim _{x\rightarrow 0+}\int _x^1g(s)\,\hspace{0.56905pt}\text{d}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

Coefficient Matrices of a Quantum Discrete Auxiliary Linear Problem

In the present paper, we describe general properties of quantum matrices that are coefficient matrices of an auxiliary problem for quantum discrete three-dimensional integrable models. Our goal is to prove a universal functional equation for the quantum determinant in the case of a finite-dimensional representation of a local Weyl algebra. Bibliography: 4 titles.

Spectral Curves and Parameterization of a Discrete Integrable Three-Dimensional Model

We consider a discrete classical integrable model on a three-dimensional cubic lattice. The solutions of this model can be used to parameterize the Boltzmann weights of various three-dimensional spin models. We find the general solution of this model constructed in terms of the theta functions defined on an arbitrary compact algebraic curve. Imposing periodic boundary conditions fixes the algebraic curve. We show that the curve then coincides with the spectral curve of the auxiliary linear problem. For a rational curve, we construct the soliton solution of the model.

Integration of the soliton hierarchy with mixed sources

We propose a method to find the explicit evolution equation for eigenfunction of the auxiliary linear problems of the soliton hierarchy with mixed sources which consist of both the sum of square of eigenfunctions and integral over the square of eigenfunctions. Then we determine the evolution equations of scattering data corresponding to the soliton hierarchy with mixed sources and solve the equation in the soliton hierarchy with mixed sources by inverse scattering transformation. The new mKdV hierarchy with mixed sources is used as an example to illustrate the method we proposed, and the nonlinear Schrödinger equation hierarchy with mixed sources, the KdV hierarchy with mixed sources are integrated. This approach can be applied to all other (1+1)-dimensional soliton hierarchies with mixed sources.

Integration of the modified Korteweg-de Vries hierarchy with an integral type of source

The modified Korteweg-de Vries hierarchy with an integral type of source (mKdVHWS), which consists of the reduced AKNS eigenvalue problem with r = q and the mKdV hierarchy with an extra term of the integration of a square eigenfunction, is investigated. We propose a method to find the explicit evolution equation for the eigenfunction of the auxiliary linear problems of the mKdVHWS. Then we determine the evolution equations of scattering data corresponding to the mKdVHWS, and solve the equation in the mKdVHWS by inverse scattering transformation.

Investigation of Newton Methods in Which Inner Iterative Processes Are Used

We investigate the conditions for convergence of iterative Newton methods, in which auxiliary linear problems are solved approximately. We propose efficient criteria for the termination of inner linear iterations.

On associativity equations

We consider the associativity or Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations and discuss their solution class based on the existence of the residue formulas, which is most relevant for nonperturbative physics. We demonstrate that for this case, proving the associativity equations reduces to solving a system of linear algebraic equations. Particular examples of solutions related to Landau–Ginzburg topological theories, Seiberg–Witten theories, and the tau functions of semiclassical hierarchies are discussed in detail. We also discuss related questions including the covariance of associativity equations, their relation to dispersionless Hirota relations, and the auxiliary linear problem for the WDVV equations.

Preconditioning by projectors in the solution of contact problems : a parallel implementation

A non-overlapping domain decomposition algorithm of the Neumann–Neumann type for solving contact problems of elasticity is presented. Using the duality theory of convex programming, the discretized problem turns into a quadratic one with equality and bound constraints. The dual problem is modified by orthogonal projectors to the natural coarse space. The resulting problem is solved by an augmented Lagrangian algorithm. The projectors ensure an optimal convergence rate for the solution of the auxiliary linear problems by the preconditioned conjugate gradient method. Relevant aspects on the numerical linear algebra of these problems are presented, together with an efficient parallel implementation of the method.

Complex of three-dimensional solvable models

An overview is given of integrable models of quantum, classical and statistical mechanics defined as evolution models in wholly discrete ( 2+1 )-dimensional space-time and based on a special type of auxiliary linear problem.

Quantum relativistic Toda chain at root of unity: Invariant approach

Recent results, concerning the quantum relativistic Toda chain at root of unity, are interpreted in this note from the point of view of invariant auxiliary linear problem. Such form of the linear problem arises in the three-dimensional integrable systems, and several results from the three dimensional models are applied to the relativistic Toda chain.

Solution of contact problems by FETI domain decomposition with natural coarse space projections

An efficient non-overlapping domain decomposition algorithm of the Neumann–Neumann type for solving both coercive and semicoercive contact problems is presented. The discretized problem is first turned by the duality theory of convex programming to the quadratic programming problem with bound and equality constraints and the latter is further modified by means of orthogonal projectors to the natural coarse space introduced by Farhat and Roux in the framework of their FETI method. The resulting problem is then solved by an augmented Lagrangian type algorithm with an outer loop for the Lagrange multipliers for the equality constraints and an inner loop for the solution of the bound constrained quadratic programming problems. The projectors are shown to guarantee fast convergence of iterative solution of auxiliary linear problems and to comply with efficient quadratic programming algorithms proposed earlier. Reported theoretical results and numerical experiments indicate high numerical scalability of the algorithm which preserves the parallelism of the FETI methods.

Duality-based domain decomposition with natural coarse-space for variational inequalities

An efficient non-overlapping domain decomposition algorithm of Neumann–Neumann type for solving variational inequalities arising from the elliptic boundary value problems with inequality boundary conditions has been presented. The discretized problem is first turned by the duality theory of convex programming into a quadratic programming problem with bound and equality constraints and the latter is further modified by means of orthogonal projectors to the natural coarse space introduced recently by Farhat and Roux. The resulting problem is then solved by an augmented Lagrangian type algorithm with an outer loop for the Lagrange multipliers for the equality constraints and an inner loop for the solution of the bound constrained quadratic programming problems. The projectors are shown to guarantee an optimal rate of convergence of iterative solution of auxiliary linear problems. Reported theoretical results and numerical experiments indicate high numerical and parallel scalability of the algorithm.

A Hierarchy of Multidimensional Hénon-Heiles Systems

A hierarchy of multidimensional Henon-Heiles (M-H-H) systems are constructed via the x- and t n -higher-order-constrained flows of KdV hierarchy. The Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV hierarchy. By using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed.