Two-phase Stefan problem with vanishing specific heat

The unique solvability of the two-phase Stefan problem with a small parameter ε ∈ [0; ε ₀] at the time derivative in the heat equations is proved. The solution is obtained on a certain time interval [0; t ₀] independent of ε. The solution of the Stefan problem is compared with the solution to the Hele–Shaw problem corresponding to the case ε = 0. The solutions of both problems are not assumed to coincide at the initial moment of time. Bibliography: 18 titles. Dedicated to Vsevolod Alekseevich Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 337–363.     

Semidiscrete and asymptotic approximations for the nonstationary radiative–conductive heat transfer problem in a periodic system of grey heat shields

We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε ^3/4) for asymptotic approximations. Bibliography: 9 titles.     

The order of convergence in the stefan problem with vanishing specific heat

The paper is concerned with the two-phase Stefan problem with a small parameter ϵ, which coresponds to the specific heat of the material. It is assumed that the initial condition does not coincide with the solution for t = 0 of the limit problem related to ε = 0. To remove this discrepancy, an auxiliary boundary layer type function is introduced. It is proved that the solution to the two-phase Stefan problem with parameter ϵ differs from the sum of the solution to the limit Hele–Shaw problem and a boundary layer type function by quantities of order O(ϵ). The estimates are obtained in H?lder norms. Bibliography: 13 titles.     

Homogenization of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain — the influence of the scaling on the nonlinearity in the effective sink-source term

In this paper, we study the asymptotic behavior of solutions u _ε of the initial boundary value problem for parabolic equations in domains W_e ì \mathbbRⁿ {\Omega_\varepsilon } \subset {\mathbb{R}^n} , n ≥ 3, perforated periodically by balls with radius of critical size ε ^α , α = n/(n − 2), and distributed with period ε. On the boundary of the balls a nonlinear third boundary condition is imposed. The weak convergence of the solutions u _ε to the solution of an effective equation is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and a corrector result with respect to the energy norm is proved.     

Justification of a quasistationary approximation for the Stefan problem

Unique solvability of the one-phase Stefan problem with a small multiplier ε at the time derivative in the equation is proved on a certain time interval independent of ε for ε ∈ (0, ε₀). The solution to the Stefan problem is compared with the solution to the Hele-Show problem, which describes the process of melting materials with zero specific heat ε and can be regarded as a quasistationary approximation for the Stefan problem. It is shown that the difference of the solutions has order . This provides a justification of the quasistationary approximation. Bibliography: 23 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 209–253.     

Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary conditions σ _ε (u _ε ) + εκ _m (u _ε ) = εg _ε ^(m) , m = 1, 2, are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is performed as ε → 0, namely, the convergence theorem for both the solution and the energy integral is proved without using an extension operator, asymptotic approximations for the solution and the energy integral are constructed, and the corresponding approximation error estimates are obtained.     

Approximations of the exponential of an operator with periodic coefficients

We consider the operator exponential e ^−tA , t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in \mathbbRⁿ {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm ||   ·   ||_{L²( \mathbbRⁿ ) ? L²( \mathbbRⁿ )} {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order O( t ^{- \fracm2} ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N _α (x), x ? \mathbbRⁿ x \in {\mathbb{R}^n} , with multi-indices α of length | a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε ^m ) in the L ²-norm as ε → 0. Bibliography: 14 titles.     

A new regularization method for a Cauchy problem of the time fractional diffusion equation

In this paper, we consider a Cauchy problem of the time fractional diffusion equation (TFDE). Such problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α (0 < α ≤ 1). We show that the Cauchy problem of TFDE is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates in the interior and on the boundary of solution domain are obtained respectively under different a-priori bound assumptions for the exact solution and suitable choices of regularization parameters. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Integer knapsack problems with set-up weights

The Integer Knapsack Problem with Set-up Weights (IKPSW) is a generalization of the classical Integer Knapsack Problem (IKP), where each item type has a set-up weight that is added to the knapsack if any copies of the item type are in the knapsack solution. The k-item IKPSW (kIKPSW) is also considered, where a cardinality constraint imposes a value k on the total number of items in the knapsack solution. IKPSW and kIKPSW have applications in the area of aviation security. This paper provides dynamic programming algorithms for each problem that produce optimal solutions in pseudo-polynomial time. Moreover, four heuristics are presented that provide approximate solutions to IKPSW and kIKPSW. For each problem, a Greedy heuristic is presented that produces solutions within a factor of 1/2 of the optimal solution value, and a fully polynomial time approximation scheme (FPTAS) is presented that produces solutions within a factor of ε of the optimal solution value. The FPTAS for IKPSW has time and space requirements of O(nlog n+n/ε ²+1/ε ³) and O(1/ε ²), respectively, and the FPTAS for kIKPSW has time and space requirements of O(kn ²/ε ³) and O(k/ε ²), respectively.     

Multidimensional two-phase quasistationary stefan problem

Existence and uniqueness of the classic solution to a two-dimensional quasistationary Stefan problem are considered. The family of model problems dependent on the parameter ε>0 which defines a heat conductivity of a matter in the direction of thex-axis is analysed. When ε→0 it is approximated by the approximate model problem having less dimensions. Analogous results are also valid for a three-dimensional problem.     

Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets

In this paper, by using probabilistic methods, we establish sharp two-sided large time estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] (i.e., for the Dirichlet heat kernels of m − (m ^2/α  − Δ) ^α/2 with m ∈ (0, 1]) in half-space-like C ^{1, 1} open sets. The estimates are uniform in m in the sense that the constants are independent of m ∈ (0, 1]. Combining with the sharp two-sided small time estimates, established in Chen et al. (Ann Probab, 2011), valid for all C ^{1, 1} open sets, we have now sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C ^{1, 1} open sets for all times. Integrating the heat kernel estimates with respect to the time variable, one can recover the sharp two-sided Green function estimates for relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C ^{1, 1} open sets established recently in Chen et al. (Stoch Process their Appl, 2011).     

Convergence theorems for solutions and energy functionals of boundary value problems in thick multilevel junctions of a new type with perturbed neumann conditions on the boundary of thin rectangles

Boundary value problems for the Poisson equation are considered in a multilevel thick junction consisting of a junction body and a lot of alternating thin rectangles of two levels depending on their lengths. Rectangles of the first level have a finite length, whereas rectangles of the second level have a length ε ^α , 0 < α < 1, where ε is the alternation period. On the boundary of thin rectangles, an inhomogeneous Neumann boundary condition involving additional perturbation parameters is imposed. We prove convergence theorems for solutions and energy integrals. Regarding the convergence of solutions of the original problem to solutions of the homogenized problem, we establish some (auxiliary) estimates necessary for obtaining the convergence rate. Bibliography: 48 titles. Illustrations: 3 figures. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 113–132.     

Scales of ε-uniform a priori estimates for nonstationary convection-dominated diffusion problems

Convection-diffusion problems of practical interest are often convection-dominated in the sense that the diffusion parameter ε satisfies 0<ε≪1. Standard error estimates for numerical approximations of such problems mostly contain constants which depend either reciprocally on ε or at least on higher order norms of the solution. These norms possibly increase to infinity in the hyperbolic limit ε→0. For the Lagrange-Galerkin scheme such error estimates have been proven by Douglas and Russell, Pironneau or Süli. In the particular case of the Navier-Stokes equations Johnson, Rannacher and Boman have observed that some of those constants which arise in existing analyses depend exponentially on the Reynolds number. Consequently, standard error analyses have no real meaning in the case of convection-dominance. Recently, Bause and Knabner have developed a rigorous ε-uniform convergence theory for finite element and Lagrange-Galerkin approximations of convection-dominated diffusion problems. The error analysis is heavily based on ε-uniform a priori estimates for the solution of the continuous problems. In their work such ε-uniform estimates are established in the Sobolev spacesL ²(Ω) andH ²(Ω) in a Lagrangian framework by transforming the convection-diffusion problem into subcharacteristic coordinates. To obtain the ε-uniformH ²-bound, strong conditions about the right-hand side which cannot realistically be assumed in practice are supposed. In this paper we demonstrate how the method of complex interpolation of Banach spaces can be employed to derive elegantly ε-uniform a priori estimates in intermediate Sobolev spaces of fractional order. We will observe that in particularH ^α-regularity of the data has to be supposed to get an ε-uniformH ^α-bound for the solution. To derive the desired estimates we use several complex interpolation results which we provide first.

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Sunto I problemi di convezione-diffusione di interesse applicativo sono spesso convezione-dominati, nel senso che il parametro di diffusione ε soddisfa 0<ε≪1. Le stime standard dell’errore nelle approssimazioni numeriche di tali problemi generalmente contengono costanti che dipendono o reciprocamente da ε o almeno da norme più elevate della soluzione. Queste norme possono tendere all’infinito nel limite iperbolico ε→0. Queste stime sono state determinate, per lo schema di Lagrange-Galerkin, da Douglas e Russell, Pironneau o Süli. Nel caso particolare delle equazioni di Navier-Stokes, Johnson, Rannacher e Boman hanno osservato che alcune di queste costanti dipendono esponenzialmente dal numero di Reynolds. Di conseguenza, l’analisi standard dell’errore non è significativa nel caso di dominanza della convezione. Recentemente, Bause e Knabner hanno sviluppato una teoria rigorosa della ε-convergenza per le approssimazioni agli elementi una teoria rigorosa della ε-convergenza per le approssimazioni agli elementi finite e per quelle di Lagrange-Galerkin relativamente a problemi di diffusione convezione-dominati. l’analisi dell’errore è pesantemente basata su stime a priori ε-uniformi per la soluzione dei problemi continui. In quel lavoro, tali stime sono stabilite negli spazi di SobolevL ²(Ω) eH ²(Ω) in rappresentazione Lagrangiana trasformando il problema di convezione-diffusione in coordinate sub-caratteristiche. Per ottenere la stima ε-uniforme inH ²(Ω), si fanno delle ipotesi stringenti sul secondo membro le quali possono non essere realisticamente assunte nella pratica. In questo lavoro mostriamo come il metodo dell’interpolazione complessa degli spazi di Banach possa essere usato per ottenere eleganti stime a priori ε-uniformi in spazi di Sobolev ad esponente frazionario. In particolare, osserveremo che per ottenere una stima ε-uniforme inH ^α per la soluzione occorre assumereH ^α regolarità sui dati. Per derivare le stime desiderate facciamo uso di diversi risultati di interpolazione complessa che vengono forniti.

     

Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ω_ε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σ_ε(u _ε) + εκ _m (u _ε) = εg ^(m) _ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure.     

Piecewise-linear,discontinuous Galerkin method for a fractional diffusion equation

We use a piecewise-linear, discontinuous Galerkin method for the time discretization of a fractional diffusion equation involving a parameter in the range − 1 < α < 0. Our analysis shows that, for a time interval (0,T) and a spatial domain Ω, the error in L_￥((0,T);L₂(W))L_\infty\bigr((0,T);L_2(\Omega)\bigr) is of order k ^2 + α , where k denotes the maximum time step. Since derivatives of the solution may be singular at t = 0, our result requires the use of non-uniform time steps. In the limiting case α = 0 we recover the known O(k ²) convergence for the classical diffusion (heat) equation. We also consider a fully-discrete scheme that employs standard (continuous) piecewise-linear finite elements in space, and show that the additional error is of order h ²log(1/k). Numerical experiments indicate that our O(k ^2 + α ) error bound is pessimistic. In practice, we observe O(k ²) convergence even for α close to − 1.     

O(dlog?N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling

In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high-dimensional applications. We describe the multifolding or quantics-based tensor approximation method of O(dlog N)-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set {1,…,N}^⊗d , or briefly N-d tensors of size N ^d , and to the respective discretized differential-integral operators in ℝ ^d . As the basic approximation result, we prove that a complex exponential sampled on an equispaced grid has quantics rank 1. Moreover, a Chebyshev polynomial, sampled over a Chebyshev Gauss–Lobatto grid, has separation rank 2 in the quantics tensor format, while for the polynomial of degree m over a Chebyshev grid the respective quantics rank is at most 2m+1. For N-d tensors generated by certain analytic functions, we give a constructive proof of the O(dlog Nlog ε ⁻¹)-complexity bound for their approximation by low-rank 2-(dlog N) quantics tensors up to the accuracy ε>0. In the case ε=O(N ^−α ), α>0, our approach leads to the quantics tensor numerical method in dimension d, with the nearly optimal asymptotic complexity O(d/αlog ² ε ⁻¹). From numerical examples presented here, we observe that the quantics tensor method has proved its value in application to various function related tensors/matrices arising in computational quantum chemistry and in the traditional finite element method/boundary element method (FEM/BEM). The tool apparently works.     

Rank-two update algorithms for the minimum volume enclosing ellipsoid problem

We consider the problem of computing a (1+ε)-approximation to the minimum volume enclosing ellipsoid (MVEE) of a given set of m points in R ⁿ . Based on the idea of sequential minimal optimization (SMO) method, we develop a rank-two update algorithm. This algorithm computes an approximate solution to the dual optimization formulation of the MVEE problem, which updates only two weights of the dual variable at each iteration. We establish that this algorithm computes a (1+ε)-approximation to MVEE in O(mn ³/ε) operations and returns a core set of size O(n ²/ε) for ε∈(0,1). In addition, we give an extension of this rank-two update algorithm. Computational experiments show the proposed algorithms are very efficient for solving large-scale problem with a high accuracy.     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. I (The main theorems and ideas)

Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ _n ^(α,β) of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w ^(a,b) σ _n ^(α,β) ‖_∞ ≦ C‖w ^(a,b) f‖_∞, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.      

Direct approach to <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates in homogenization theory

We derive interior L ^p -estimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic coefficients. Our approach relies on BMO-estimates and an interpolation argument, gradients are treated with the help of finite differences. This allows to treat coefficients that depend on a fast and a slow variable. The estimates imply an L ^p -corrector result for approximate solutions.      

Homogenization with corrector for a multidimensional periodic elliptic operator near an edge of an inner gap

For a second order differential operator A \mathcal{A} _ε  = −div g(x/ε)∇ + ε ⁻²p(x/ε) in L ₂(ℝ ^d ) with periodic coefficients and small parameter ε > 0 we study an approximation of the resolvent of A \mathcal{A} _ε at a point close to an edge of an inner gap in the spectrum of A \mathcal{A} _ε . Under certain regularity conditions, we construct an approximation (with a first order corrector taken into account) for the resolvent with error estimate of order O(ε ²). We show that a proper effective operator and a proper corrector are associated to each (regular) edge of the gap. Bibliography: 14 titles.