Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions

In Voller, Swenson and Paola [V.R. Voller, J.B. Swenson, C. Paola, An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat Mass Transfer 47 (2004) 5387-5390], and Lorenzo-Trueba and Voller [J. Lorenzo-Trueba, V.R. Voller, Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation, J. Math. Anal. Appl. 366 (2010) 538-549], a model associated with the formation of sedimentary ocean deltas is studied through a one-phase Stefan-like problem with variable latent heat. Motivated by these works, we consider a two-phase Stefan problem with variable latent of fusion and initial temperature, and constant heat flux boundary conditions. We obtain the sufficient condition on the data in order to have an explicit solution of a similarity type of the corresponding free boundary problem for a semi-infinite material. Moreover, the explicit solution given in the first quoted paper can be recovered for a particular case by taking a null heat flux condition at the infinity.     

A posteriori error analysis for elliptic pdes on domains with complicated structures

Summary. The discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores. The model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh-width at most and mixed boundary conditions for the Poisson equation. Reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements. Emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimates. Mathematics Subject Classification (2000):65N30, 65R20, 73C50Received: 28, June 2001     

On the hydrodynamic motion in a domain with mixed boundary conditions: Existence,uniqueness, stability and linearization principle

Summary Existence, uniqueness, stability and instability theorems of hydrodynamic motion in a bounded domain with mixed (free boundary type) conditions are proved. Moreover a linearization principle is proved for an unperturbed periodic motion.Work performed under auspices of G.N.F.M.-C.N.R., also M.P.I. (40%) 20120201/81.     

Behaviour of mushy regions under the action of a volumetric heat source

In this paper we consider solutions to Stefan problems in spatial dimensions N ? 1. We find the necessary conditions on the heat source for the appearance of a ‘mushy region’ (i.e. a region where the temperature coincides identically with the temperature of the change of phase) inside a purely liquid (or solid) phase. For sources depending on energy such conditions are connected only with the local behaviour of the source near the energy level corresponding to the beginning of the change or phase. Both weak and smooth solutions are considered; in the latter case the behaviour of the solution at the free boundary is investigated in detail.     

Representation of the solution to a model problem in semiconductor physics

We study a mixed type problem for the Poisson equation arising in the modeling of charge transport in semiconductor devices [V. Romano, 2D simulation of a silicon MESFET with a non-parabolic hydrodynamical model based on the maximum entropy principle, J. Comput. Phys. 176 (2002) 70-92; A.M. Blokhin, R.S. Bushmanov, A.S. Rudometova, V. Romano, Linear asymptotic stability of the equilibrium state for the 2D MEP hydrodynamical model of charge transport in semiconductors, Nonlinear Anal. 65 (2006) 1018-1038]. Unlike well-studied elliptic boundary-value problems in domains with smooth boundaries (see, for example, [O.A. Ladyzhenskaya, N.N. Uralceva, Linear and Quasilinear Elliptic Equations, Nauka, Moscow, 1973; D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983]), our problem has two significant features: firstly, the boundary is not a smooth curve and, secondly, the type of boundary conditions is mixed (the Dirichlet condition is satisfied on the one part of the boundary whereas the Neumann condition on the other part). The well-posedness of the problem in Hölder and Sobolev spaces is proved. The representation of the solution to the problem is obtained in an explicit form.     

Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

In this work, we consider the heat equation coupled with Stokes equations under threshold type boundary condition. The conditions for existence and uniqueness of the weak solution are made clear. Next we formulate the finite element problem, recall the conditions of its solvability, and study its convergence by making use of Babuska–Brezzi's conditions for mixed problems. Third we formulate an Uzawa's type iterative algorithm that separates the fluid from heat conduction, study its feasibility, and convergence. Finally the theoretical findings are validated by numerical simulations.     

Thermally coupled,stationary, incompressible MHD flow; existence,uniqueness, and finite element approximation

This article addresses the questions of existence, uniqueness, and finite element approximation (including some computational aspects) of solutions to the equations of steady-state magnetohy-drodynamic (MHD) when buoyancy effects due to temperature differences in the flow cannot be neglected. We couple the MHD equations to the heat equation and employ the well-known Boussinesq approximation. We consider the equations posed on a bounded three-dimensional domain. The boundary conditions for the velocity are of Dirichlet type; the boundary conditions for the temperature are mixed (of Dirichlet type and of Neumann type); we also specify the normal component of the magnetic field and tangential component of the electric field on the boundary. We point out that these problems are relevant to many physical phenomena such as the cooling of nuclear reactors by electrically conducting fluids, continuous metal casting, crystal growth, and semi-conductor manufacture. © 1995 John Wiley & Sons, Inc.     

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

     

Two-dimensional steady-state oscillation problems of anisotropic elasticity

The paper deals with the two-dimensional exterior boundary value problems of the steady-state oscillation theory for anisotropic elastic bodies. By means of the limiting absorption principle the fundamental matrix of the oscillation equations is constructed and the generalized radiation conditions of Sommerfeld-Kupradze type are established. Uniqueness theorems of the basic and mixed type boundary value problems are proved.     

Sharp regularity theory for thermo-elastic mixed problems 1

We give a sharp (optimal) regularity theory of thermo-elastic mixed problems. Our approach is by P.D.E. methods and applies to any space dimension and, in principle, to any set of boundary conditions. We consider two sets of boundary conditions: hinged and clamped B.C. The original coupled P.D.E. system is split into two suitable uncoupled P.D.E. equations: a Kirchoff mixed problem and a heat equation, whose delicate, optimal regularity is available in the literature. Ultimately, the original problem with boundary non-homogeneous term is reduced to the same problem, however, with homogeneou B.C. and a known ‘right-hand term’ in the equation, which is easier to analyze.     

Dual finite element analysis for some elliptic variational equations and inequalities

In some boundary-value problems the gradient or the cogradient of the solution is more important than the solution itself. Dual variational formulation of elliptic problems is utilized to define finiteelement approximations of the cogradient. A priori error estimates are presented for a class of second-order elliptic problems, including problems of elastostatics. If the boundary conditions are classical (i.e., of Dirichlet, Neumann. Newton, or mixed type), the primal and dual formulations are equivalent with variational equations, whereas the unilateral boundary conditions lead to variational inequalities. The paper has a surveyable character.     

Approximation par la méthode des éléments finis de la formulation en domaine régulier de problèmes de fissures

The discretization by various mixed finite element methods of a new variational formulation of crack problems is considered. The new formulation, called the smooth domain method, is derived for crack problems in the case of a simplified model of an elastic membrane. Inequality type boundary conditions are prescribed at the crack faces. The resulting model takes the form of an unilateral contact problem on the crack. The mathematical analysis for the method leads to optimal convergence rates, as given in this Note. To cite this article: Z. Belhachmi et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The existence of approximate solutions to mixed boundary value problems

Approximate solutions, similar to the type used in the Complex Variable Boundary Element Method, are shown to exist for two dimensional mixed boundary value potential problems on multiply connected domains. These approximate solutions can be used numerically to obtain least squares solutions or solutions which interpolate given boundary conditions. Areas of application include fluid flow around obstacles and heat flow in a domain with insulated boundary segments. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 191–199, 1999     

On Jacobi’s condition for the simplest problem of calculus of variations with mixed boundary conditions

The purpose of this paper is an extension of Jacobi’s criteria for positive definiteness of second variation of the simplest problems of calculus of variations subject to mixed boundary conditions. Both non constrained and isoperimetric problems are discussed. The main result is that if we stipulate conditions (21) and (22) then Jacobi’s condition remains valid also for the mixed boundary conditions.     

Dual extremum principles and error bounds for a class of boundary value problems

Error bounds for a wide class of linear and nonlinear boundary value problems are derived from the theory of dual extremum principles. The results are illustrated by two examples arising in the theory of heat transfer, which involve mixed boundary conditions.     

A theoretical and numerical approach to some free boundary problems

Summary In this paper some free boundary problems related to the flow of fluids in porous media are studied. Using a method due to Baiocchi, for these problems we not only establish theoretical results (existence and uniqueness theorems for the solution) but at the same time develop an algorithm for the numerical approach of the solution. Such an algorithm is rigorous from a mathematical point of view and it competes very well with the ones already known both in simplicity of programming and in speed of execution. Entrata in Redazione il 18 luglio 1973. ? Laboratorio di Analisi Numerica del C.N.R. di Pavia ? and ? Università di Pavia ?. This work was supported by C.N.R. in the frame of L.A.N. at Pavia.     

A solution to an open problem for wave equations with generalized Wentzell boundary conditions

In this paper we solve an open problem put forward by A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, concerning the mixed problem for wave equations with generalized Wentzell boundary conditions. As a consequence, we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized Wentzell boundary conditions. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, Tübingen, Germany (e-mail: tixi@fa.uni-tuebingen.de;jili@fa.uni-tuebingen.de)The first author acknowledges support from the Alexander-von-Humboldt Foundation and from CAS and NSFC. The second author acknowledges support from the Max-Planck Society and from CAS and EMC.     

Free boundary problems in the theory of fluid flow through porous media: Existence and uniqueness theorems

Summary Elliptic free boundary problems in the theory of fluid flow through porous media are studied by a new method, which reduces the problems to variational inequalities: existence and uniqueness theorems are proved. Entrata in Redazione il 3 agosto 1972. Research supported by C.N.R. in the frame of the collaboration between L.A.N. of Pavia and E.R.A. 215 of C.N.R.S. and of Paris University. ? Laboratorio di Analisi Numerica del C.N.R. di Pavia ? and ? Università di Pavia ?. ? Università di Pavia ? and ? G.N.A.F.A. del C.N.R. ?.     

Boundary characteristic point regularity for semilinear reaction-diffusion equations: towards an ode criterion

The classical problem of regularity of boundary characteristic points for semilinear heat equations with homogeneous Dirichlet conditions is considered. The Petrovskii ( 2?{loglog} ) \left( {2\sqrt {{\log \log }} } \right) criterion (1934) of the boundary regularity for the heat equation can be adapted to classes of semilinear parabolic equations of reaction–diffusion type and takes the form of an ordinary differential equation (ODE) regularity criterion. Namely, after a special matching with a boundary layer, the regularity problem reduces to a onedimensional perturbed nonlinear dynamical system for the first Fourier-like coefficient of the solution in an inner region. A similar ODE criterion, with an analogous matching procedures, is shown formally to exist for semilinear fourth order biharmonic equations of reaction-diffusion type. Extensions to regularity problems of backward paraboloid vertices in \mathbbR^N {\mathbb{R}^N} are discussed. Bibliography: 54 titles. Illustrations: 1 figure.     

Application of generalized separation of variables to solving mixed problems with irregular boundary conditions

One of the methods for solving mixed problems is the classical separation of variables (the Fourier method). If the boundary conditions of the mixed problem are irregular, this method, generally speaking, is not applicable. In the present paper, a generalized separation of variables and a way of application of this method to solving some mixed problems with irregular boundary conditions are proposed. Analytical representation of the solution to this irregular mixed problem is obtained.