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## Dirichlet Boundary Conditions (dirichlet + boundary_condition)
Kinds of Dirichlet Boundary Conditions
## Selected Abstracts## A finite element-based level set method for structural optimization INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2010Xianghua XingAbstract A finite element-based level set method is implemented for structural optimization. The streamline diffusion finite element method is used for solving both the level set equation and the reinitialization equation. The lumped scheme is addressed and the accuracy is compared with the conventional finite difference-based level set method. A Dirichlet boundary condition is enforced during the reinitialization to prevent the boundary from drifting. Numerical examples of minimum mean compliance design illustrate the reliability of the proposed optimization method. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## A Petrov,Galerkin finite element model for one-dimensional fully non-linear and weakly dispersive wave propagation INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2001Seung-Buhm WooAbstract A new finite element method is presented to solve one-dimensional depth-integrated equations for fully non-linear and weakly dispersive waves. For spatial integration, the Petrov,Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C2 -continuity. For the time integration an implicit predictor,corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth-order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Non-homogeneous Navier,Stokes systems with order-parameter-dependent stresses MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2010Helmut AbelsAbstract We consider the Navier,Stokes system with variable density and variable viscosity coupled to a transport equation for an order-parameter c. Moreover, an extra stress depending on c and ,c, which describes surface tension like effects, is included in the Navier,Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two-phase flow of viscous incompressible fluids. The so-called density-dependent Navier,Stokes system is also a special case of our system. We prove short-time existence of strong solution in Lq -Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## On an initial-boundary value problem for a wide-angle parabolic equation in a waveguide with a variable bottom MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2009V. A. DougalisAbstract We consider the third-order Claerbout-type wide-angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range-dependent topography. There is strong indication that the initial-boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well-posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second-order boundary condition, derived by assuming that the standard (narrow-angle) PE holds on B, and establish a priori H2 estimates for the solution of the resulting initial-boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second-order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge-type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson,Kreiss boundary condition in the wide-angle case. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Homogenization of elliptic problems with the Dirichlet and Neumann conditions imposed on varying subsets MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2007Carmen Calvo-JuradoAbstract We study the asymptotic behaviour of the solution un of a linear elliptic equation posed in a fixed domain ,. The solution un is assumed to satisfy a Dirichlet boundary condition on ,n, where ,n is an arbitrary sequence of subsets of ,,, and a Neumman boundary condition on the remainder of ,,. We obtain a representation of the limit problem which is stable by homogenization and where it appears a generalized Fourier boundary condition. We also prove a corrector result. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Global existence and uniform stability of solutions for a quasilinear viscoelastic problem MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2007Salim A. MessaoudiAbstract In this paper the nonlinear viscoelastic wave equation in canonical form with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Scattering from infinite rough tubular surfaces MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2007Xavier ClaeysAbstract We study the Helmholtz equation in the exterior of an infinite perturbed cylinder with a Dirichlet boundary condition. Existence and uniqueness of solutions are established using the variational technique introduced (SIAM J. Math. Anal. 2005; 37(2):598,618). We also provide stability estimates with explicit dependence of the constants in terms of the frequency and the perturbed cylinder thickness. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## On non-stationary viscous incompressible flow through a cascade of profiles MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2006Miloslav FeistauerAbstract The paper deals with theoretical analysis of non-stationary incompressible flow through a cascade of profiles. The initial-boundary value problem for the Navier,Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Global and blow-up solutions for non-linear degenerate parabolic systems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003Zhi-wen DuanAbstract In this paper the degenerate parabolic system ut=u(uxx+av). vt=v(vxx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (,1=,2) of solution are analysed. For , the blow-up time, blow-up rate and blow-up set of blow-up solution are estimated and the asymptotic behaviour of solution near the blow-up time is discussed by using the ,energy' method. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Time asymptotics for the polyharmonic wave equation in waveguides MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2003P. H. LeskyAbstract Let , denote an unbounded domain in ,n having the form ,=,l×D with bounded cross-section D,,n,l, and let m,, be fixed. This article considers solutions u to the scalar wave equation ,u(t,x) +(,,)mu(t,x) = f(x)e,i,t satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t,, is investigated. Depending on the choice of f ,, and ,, two cases occur: Either u shows resonance, which means that ,u(t,x),,, as t,, for almost every x , ,, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## On parallel solution of linear elasticity problems. NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2002Part II: Methods, some computer experimentsAbstract This is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method. In the first part of the trilogy block- diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg-method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M -matrices, i.e. in order to guarantee the existence of the MIC(0) factorization. In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block-diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)-factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system. In the third part of the trilogy, we will focus on the use of higher-order finite elements. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## The boundary element method with Lagrangian multipliers, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2009Gabriel N. GaticaAbstract On open surfaces, the energy space of hypersingular operators is a fractional order Sobolev space of order 1/2 with homogeneous Dirichlet boundary condition (along the boundary curve of the surface) in a weak sense. We introduce a boundary element Galerkin method where this boundary condition is incorporated via the use of a Lagrangian multiplier. We prove the quasi-optimal convergence of this method (it is slightly inferior to the standard conforming method) and underline the theory by a numerical experiment. The approach presented in this article is not meant to be a competitive alternative to the conforming method but rather the basis for nonconforming techniques like the mortar method, to be developed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] ## Lagrange interpolation and finite element superconvergence, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004Bo LiAbstract We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d -dimensional Qk -type elements with d , 1 and k , 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H1 norm. For d -dimensional Pk -type elements, we consider the standard Lagrange interpolation,the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d , 2 and k , d + 1 that such interpolation and the finite element solution are not superclose in both H1 and L2 norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33,59, 2004. [source] ## Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2006Alexander BarnettThe quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak is that every eigenfunction ,n of the Laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue En , ,); that is, "strong scars" are absent. We study numerically the rate of equidistribution for a uniformly hyperbolic, Sinai-type, planar Euclidean billiard with Dirichlet boundary condition (the "drum problem") at unprecedented high E and statistical accuracy, via the matrix elements ,,n, Â,m, of a piecewise-constant test function A. By collecting 30,000 diagonal elements (up to level n , 7 × 105) we find that their variance decays with eigenvalue as a power 0.48 ± 0.01, close to the semiclassical estimate ½ of Feingold and Peres. This contrasts with the results of existing studies, which have been limited to En a factor 102 smaller. We find strong evidence for QUE in this system. We also compare off-diagonal variance as a function of distance from the diagonal, against Feingold-Peres (or spectral measure) at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions. © 2006 Wiley Periodicals, Inc. [source] ## He's homotopy perturbation method for two-dimensional heat conduction equation: Comparison with finite element method HEAT TRANSFER - ASIAN RESEARCH (FORMERLY HEAT TRANSFER-JAPANESE RESEARCH), Issue 4 2010M. JalaalAbstract Heat conduction appears in almost all natural and industrial processes. In the current study, a two-dimensional heat conduction equation with different complex Dirichlet boundary conditions has been studied. An analytical solution for the temperature distribution and gradient is derived using the homotopy perturbation method (HPM). Unlike most of previous studies in the field of analytical solution with homotopy-based methods which investigate the ODEs, we focus on the partial differential equation (PDE). Employing the Taylor series, the gained series has been converted to an exact expression describing the temperature distribution in the computational domain. Problems were also solved numerically employing the finite element method (FEM). Analytical and numerical results were compared with each other and excellent agreement was obtained. The present investigation shows the effectiveness of the HPM for the solution of PDEs and represents an exact solution for a practical problem. The mathematical procedure proves that the present mathematical method is much simpler than other analytical techniques due to using a combination of homotopy analysis and classic perturbation method. The current mathematical solution can be used in further analytical and numerical surveys as well as related natural and industrial applications even with complex boundary conditions as a simple accurate technique. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20292 [source] ## Total FETI,an easier implementable variant of the FETI method for numerical solution of elliptic PDE INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2006k DostálAbstract A new variant of the FETI method for numerical solution of elliptic PDE is presented. The basic idea is to simplify inversion of the stiffness matrices of subdomains by using Lagrange multipliers not only for gluing the subdomains along the auxiliary interfaces, but also for implementation of the Dirichlet boundary conditions. Results of numerical experiments are presented which indicate that the new method may be even more efficient then the original FETI. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## The use of negative penalty functions in solving partial differential equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2005Sinniah IlankoAbstract In variational and optimization problems where the field variable is represented by a series of functions that individually do not satisfy the constraints, penalty functions are often used to enforce the constraint conditions approximately. The major drawback with this approach is that the error due to any violation of the constraint is not known. In a recent publication dealing with the Rayleigh,Ritz method it was shown that, by using a combination of positive and negative penalty parameters, any error due to the violation of the constraints may be kept within any desired tolerance. This paper shows that this approach may also be used in solving partial differential equations using a Galerkin's solution to Laplace's equation subject to mixed Neumann and Dirichlet boundary conditions as an example. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Solution of two-dimensional Poisson problems in quadrilateral domains using transfinite Coons interpolation INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2004Christopher G. ProvatidisAbstract This paper proposes a global approximation method to solve elliptic boundary value Poisson problems in arbitrary shaped 2-D domains. Using transfinite interpolation, a symmetric finite element formulation is derived for degrees of freedom arranged mostly along the boundary of the domain. In cases where both Dirichlet and Neumann boundary conditions occur, the numerical solution is based on bivariate Coons interpolation using the boundary only. Furthermore, in case of only Dirichlet boundary conditions and no existing axes of symmetry, it is proposed to use at least one internal point and apply transfinite interpolation. The theory is sustained by five numerical examples applied to domains of square, circular and elliptic shape. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Approximate imposition of boundary conditions in immersed boundary methods INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2009Ramon CodinaAbstract We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non-symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Imposing Dirichlet boundary conditions in the extended finite element method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2006Nicolas MoësAbstract This paper is devoted to the imposition of Dirichlet-type conditions within the extended finite element method (X-FEM). This method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces. Imposing Neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh (observed numerically in earlier papers). On the contrary, much less work has been devoted to Dirichlet boundary conditions for the X-FEM (or the limiting case of stiff boundary conditions). In this paper, we introduce a strategy to impose Dirichlet boundary conditions while preserving the optimal rate of convergence. The key aspect is the construction of the correct Lagrange multiplier space on the boundary. As an application, we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding (RTM) process while avoiding remeshing. The case of inner conditions is also discussed as well as two important practical cases: material interfaces and phase-transformation front capturing. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003N. SukumarArticle first published online: 11 MAR 200Abstract Voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids. Natural neighbours are based on the Voronoi diagram, which partitions space into closest-point regions. The Sibson and the Laplace (non-Sibsonian) interpolants which are based on natural neighbours have shown promise within a Galerkin framework for the solution of partial differential equations. In this paper, we focus on the Laplace interpolant with a two-fold objective: first, to unify the previous developments related to the Laplace interpolant and to indicate its ties to some well-known numerical methods; and secondly to propose a Voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids. A conservation law in integral form is discretized on Voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids. The proposed scheme can also be viewed as a point collocation technique. A detailed study on consistency is conducted, and the satisfaction of the discrete maximum principle (stability) is established. Owing to symmetry of the Laplace weight, a symmetric positive-definite stiffness matrix is realized which permits the use of efficient linear solvers. On a regular (rectangular or hexagonal) grid, the difference scheme reduces to the classical finite difference method. Numerical examples for the Poisson equation with Dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Geometrical interpretation of the multi-point flux approximation L-method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2009Yufei CaoAbstract In this paper, we first investigate the influence of different Dirichlet boundary discretizations on the convergence rate of the multi-point flux approximation (MPFA) L-method by the numerical comparisons between the MPFA O- and L-method, and show how important it is for this new method to handle Dirichlet boundary conditions in a suitable way. A new Dirichlet boundary strategy is proposed, which in some sense can well recover the superconvergence rate of the normal velocity. In the second part of the work, the MPFA L-method with homogeneous media is studied. A systematic concept and geometrical interpretations of the L-method are given and illustrated, which yield more insight into the L-method. Finally, we apply the MPFA L-method for two-phase flow in porous media on different quadrilateral grids and compare its numerical results for the pressure and saturation with the results of the two-point flux approximation method. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## The natural volume method (NVM): Presentation and application to shallow water inviscid flows INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2009R. AtaAbstract In this paper a fully Lagrangian formulation is used to simulate 2D shallow water inviscid flows. The natural element method (NEM), which has been used successfully with several solid and fluid mechanics applications, is used to approximate the fluxes over Voronoi cells. This particle-based method has shown huge potential in terms of handling problems involving large deformations. Its main advantage lies in the interpolant character of its shape function and consequently the ease it allows with respect to the imposition of Dirichlet boundary conditions. In this paper, we use the NEM collocationally, and in a Lagrangian kinematic description, in order to simulate shallow water flows that are boundary moving problems. This formulation is ultimately shown to constitute a finite-volume methodology requiring a flux computation on Voronoi cells rather than the standard elements, in a triangular or quadrilateral mesh. St Venant equations are used as the mathematical model. These equations have discontinuous solutions that physically represent the existence of shock waves, meaning that stabilization issues have thus been considered. An artificial viscosity deduced from an analogy with Riemann solvers is introduced to upwind the scheme and therefore stabilize the method. Some inviscid bidimensional flows were used as preliminary benchmark tests, which produced decent results, leading to well-founded hopes for the future of this method in real applications. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Optimal flow control for Navier,Stokes equations: drag minimization INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2007L. DedèAbstract Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag minimization for a body in relative motion in a fluid by controlling the velocity through the body boundary. With this aim, we handle with an optimal control approach applied to the steady incompressible Navier,Stokes equations. We use the Lagrangian functional approach and we consider the Lagrangian multiplier method for the treatment of the Dirichlet boundary conditions, which include the control function itself. Moreover, we express the drag coefficient, which is the functional to be minimized, through the variational form of the Navier,Stokes equations. In this way, we can derive, in a straightforward manner, the adjoint and sensitivity equations associated with the optimal control problem, even in the presence of Dirichlet control functions. The problem is solved numerically by an iterative optimization procedure applied to state and adjoint PDEs which we approximate by the finite element method. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2009Aissa GuesmiaAbstract In this paper we consider the following Timoshenko system: with Dirichlet boundary conditions and initial data where a, b, g and h are specific functions and ,1, ,2, k1, k2 and L are given positive constants. We establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on h at the origin, we show that the energy of the system is bounded above by a quantity, depending on g and h, which tends to zero as time goes to infinity. Our estimate allows us to consider a large class of functions h with general growth at the origin. We use some examples (known in the case of wave equations and Maxwell system) to show how to derive from our general estimate the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize some existing results in the literature and generate some interesting open problems. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## On a Penrose,Fife type system with Dirichlet boundary conditions for temperature MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2003Gianni GilardiWe deal with the Dirichlet problem for a class of Penrose,Fife phase field models for phase transitions. An existence result is obtained by approximating the non-homogeneous Dirichlet condition with classical third type conditions on the heat flux at the boundary of the domain where the model is considered. Moreover, we prove a regularity and uniqueness result under stronger assumptions on the regularity of the data. Suitable assumptions on the behaviour of the heat flux at zero and +,are considered. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Global existence of solutions for non-small data to non-linear spherically symmetric thermoviscoelasticity MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2003J. GawineckiAbstract We consider some initial,boundary value problems for non-linear equations of thermoviscoelasticity in the three-dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Note on a versatile Liapunov functional: applicability to an elliptic equation MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2002J. N. FlavinA novel, very effective Liapunov functional was used in previous papers to derive decay and asymptotic stability estimates (with respect to time) in a variety of thermal and thermo-mechanical contexts. The purpose of this note is to show that the versatility of this functional extends to certain non-linear elliptic boundary value problems in a right cylinder, the axial co-ordinate in this context replacing the time variable in the previous one. A steady-state temperature problem is considered with Dirichlet boundary conditions, the condition on the boundary being independent of the axial co-ordinate. The functional is used to obtain an estimate of the error committed in approximating the temperature field by the two-dimensional field induced by the boundary condition on the lateral surface. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains MATHEMATISCHE NACHRICHTEN, Issue 12 2008Annegret GlitzkyAbstract We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain ,0 of the domain of definition , of the energy balance equation and of the Poisson equation. Here ,0 corresponds to the region of semiconducting material, , \ ,0 represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a W1,p -regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Some remarks on essential self-adjointness and ultracontractivity of a class of singular elliptic operators MATHEMATISCHE NACHRICHTEN, Issue 8 2008Michael M. H. PangAbstract We study the properties of essential self-adjointness on C,c (,N) and semigroup ultracontractivity of a class of singular second order elliptic operators defined in L2 (,N, ,,a ,N(x) dx) with Dirichlet boundary conditions, where a, b , , and ,: ,N , (0, ,) is a C, -function satisfying c -1(1 + |x |) , , (x) , c (1 + |x |) (x , ,N). We also obtain sharp short time upper and lower diagonal bounds on the heat kernel of e ,Ht. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] |