We derive a posteriori error estimates in natural energy norms with weights. They are useful in localizing the error extraction in regions of interest, and form the basis of an adaptive procedure. We use them to study the evolution of a... more
We derive a posteriori error estimates in natural energy norms with weights. They are useful in localizing the error extraction in regions of interest, and form the basis of an adaptive procedure. We use them to study the evolution of a persistent corner singularity and elucidate the issue of critical angle for instantaneous smoothing. 1 Introduction The presence of interfaces, and associated lack of regularity, is responsible for global numerical pollution effects for parabolic free boundary problems. A cure consists of equidistributing discretization errors in adequate norms by means of highly graded meshes and varying time steps. Their construction relies on a posteriori error estimates, which are a fundamental component for the design of reliable and efficient adaptive algorithms for PDEs. These issues have been recently tackled in [6], [7], [8], [9], and are briefly discussed here. We consider for simplicity the classical two-phase Stefan problem for an ideal material with consta...
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When a quiescent molten polymer is cooled below the equilibrium melting temperature, crystals (spherulites) appear and keep growing as long as the temperature ranges between the melting temperature and the glass transition temperature.... more
When a quiescent molten polymer is cooled below the equilibrium melting temperature, crystals (spherulites) appear and keep growing as long as the temperature ranges between the melting temperature and the glass transition temperature. The crystallization process depends upon temperature and crystalline microstructure. In particular, the reduction of the free volume and subsequent impingement between crystals influence both nucleation and growth rates of spherulites. Below the glass transition temperature, the polymer consists of crystal and amorphous phases. In the sequel we discuss a mathematical model for a bidimensional isothermal crystallization process which takes into account the nucleation, growth, and impingement of spherulites
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Mesh adaptation is discussed for the two-phase Stefan problem in 2-D. Three local parameters are used to equidistribute interpolation errors in maximum norm for temperature as well as to specify the so-called refined region. An extra... more
Mesh adaptation is discussed for the two-phase Stefan problem in 2-D. Three local parameters are used to equidistribute interpolation errors in maximum norm for temperature as well as to specify the so-called refined region. An extra parameter is utilized, in the event of mushy regions, to equidistribute L 1-interpolation errors for enthalpy within the mush. Upon failure of certain quality mesh tests, the current mesh is discarded and a new one completely regenerated; consecutive meshes are thus noncompatible. A typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas it is locally refined in the vicinity of the discrete interface for the relation to become hyperbolic. A drastic reduction of spatial degrees of freedom is obtained with these highly graded meshes. A suitable interpolation theory for noncompatible meshes quantifies the error introduced by mesh changes and leads to the mesh selection algorithm. The resulting scheme is stable in various Sobolev norms and convergent with an a priori prescribed rate. Binary search techniques on suitable quadtree structured data are used to reach a quasi-optimal computational complexity in several search operations necessary for both mesh generation and interpolation between noncompatible meshes. Several numerical experiments illustrate the superior performance of this method as well as its efficiency in approximating both solutions and interfaces in maximum norm.
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During recent years, degenerate parabolic equations have attracted the attention of both scientists and engineers mainly because of their relevance in modelling industrial processes, [3]. A common feature in dealing with such problems is... more
During recent years, degenerate parabolic equations have attracted the attention of both scientists and engineers mainly because of their relevance in modelling industrial processes, [3]. A common feature in dealing with such problems is the intrinsic lack of regularity of solutions across the free boundaries which, in turn, are not known in advance. For the two-phase Stefan problem, for instance, the temperature cannot be better than Lipschitz continuous and the enthalpy typically exhibits a jump discontinuity across the interface. This lack of smoothness makes piecewise linear finite elements, defined on quasi-uniform meshes, perform worse than expected according to the interpolation theory. Methods studied so far are not completely satisfactory in that they do not profit from the fact that singularities occur in a small part of the entire domain of definition, at least whenever the interface is sufficiently smooth. Consequently, a possible remedy is to be found in terms of a suitably designed adaptive algorithm. In fact, one would like to use a finer mesh near singularities in order to equidistribute the errors but still preserve the number of degrees of freedom, and thus the computational complexity. We refer to [1,6] for an account of the state-of-the-art on this topic along with numerous references.
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Research Interests: Mathematics and Curvature
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We analyse an axisymmetric mathematical model which describes the thermal effects during cementation of a femoral prosthesis. Various numerical simulations illustrate some critical aspects of this implant such as the thermal bone necrosis... more
We analyse an axisymmetric mathematical model which describes the thermal effects during cementation of a femoral prosthesis. Various numerical simulations illustrate some critical aspects of this implant such as the thermal bone necrosis and the presence of unreacted residual monomer.
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Abstract. A double obstacle problem for a singularly perturbed reaction-diffusion equation is used to approximate curvature-dependent evolving interfaces. The solution exhibits a rapid variation from-1 to 1 within a thin transition layer,... more
Abstract. A double obstacle problem for a singularly perturbed reaction-diffusion equation is used to approximate curvature-dependent evolving interfaces. The solution exhibits a rapid variation from-1 to 1 within a thin transition layer, and coincides with the obstacle±1 ...
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Research Interests: Mathematics and Semigroup
We outline the main contributions of Prof. Enrico Magenes to the analysis and numerical approximation of mathematical models of phase transition processes. Starting from the 1980’s, a semigroup approach to Stefan problems, optimal rates... more
We outline the main contributions of Prof. Enrico Magenes to the analysis and numerical approximation of mathematical models of phase transition processes. Starting from the 1980’s, a semigroup approach to Stefan problems, optimal rates of convergence for the nonlinear Chernoff formula, regularity properties of solutions, theoretical and numerical aspects of Stefan models in a concentrated capacity, were investigated by Enrico Magenes. His expertise was fundamental for developing numerical analysis of evolutionary free boundary problems and applications in a modern framework.
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Thank you very much for reading analysis and numerics of partial differential equations. Maybe you have knowledge that, people have look hundreds times for their chosen books like this analysis and numerics of partial differential... more
Thank you very much for reading analysis and numerics of partial differential equations. Maybe you have knowledge that, people have look hundreds times for their chosen books like this analysis and numerics of partial differential equations, but end up in infectious downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they cope with some infectious virus inside their laptop.
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A FULLY DISCRETE ADAPTIVE NONLINEAR CHERNOFF FORMULA* RH NOCHETTOt, M. PAOLINIt, AND C. VERDIt? Abstract. An adaptive piecewise linear finite element approximation to a linear method, the so-called non-linear Chernoff formula, for the... more
A FULLY DISCRETE ADAPTIVE NONLINEAR CHERNOFF FORMULA* RH NOCHETTOt, M. PAOLINIt, AND C. VERDIt? Abstract. An adaptive piecewise linear finite element approximation to a linear method, the so-called non-linear Chernoff formula, for the simplest two-phase ...
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Abstract. A class of multidimensional degenerate parabolic equations is considered: the two-phase Stefan problem and the porous medium equation are analyzed as models of singular parabolic equations; nonstationary filtration (with... more
Abstract. A class of multidimensional degenerate parabolic equations is considered: the two-phase Stefan problem and the porous medium equation are analyzed as models of singular parabolic equations; nonstationary filtration (with gravity) is also treated as a model of ...
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Abstract. The two-phase Stefan problem in several space dimensions and with an enthalpy dependent source is studied here. Assuming that a strictly positive temperature is prescribed at the fixed boundary, the existence of a solution with... more
Abstract. The two-phase Stefan problem in several space dimensions and with an enthalpy dependent source is studied here. Assuming that a strictly positive temperature is prescribed at the fixed boundary, the existence of a solution with an extra space regularity is proven. The stability ...
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494 ROUBÍСЕК AND VERDI 12. RH Nochetto, C. Verdi, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal. 25 (1988), 784-814. 13. T, Roubieek, Optimal control of a Stefan problem with state-space... more
494 ROUBÍСЕК AND VERDI 12. RH Nochetto, C. Verdi, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal. 25 (1988), 784-814. 13. T, Roubieek, Optimal control of a Stefan problem with state-space constraints. Numer, Math, ...
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A fully discrete scheme for a class of multidimensional degen-erate parabolic equations is proposed. The discretization is given by t!? piecewise linear finite elements in space and back-ward differences in time (the smoothing procedure... more
A fully discrete scheme for a class of multidimensional degen-erate parabolic equations is proposed. The discretization is given by t!? piecewise linear finite elements in space and back-ward differences in time (the smoothing procedure is avoided). Numerical ...
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This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a... more
This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of O ( h 1 / 2 ) O({h^{1/2}}) is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.
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Research Interests: Mathematics, Applied Mathematics, Mesh generation, Numerical Analysis, Finite element method, and 10 moreFinite Elements, Triangulation, Discretization, Numerical Analysis and Computational Mathematics, Boolean Satisfiability, Adaptive Finite Element Method, Degree of Freedom, Laplace operator, Rate of Convergence, and Sobolev space
We consider a mathematical model for solidification of semicrystalline polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. In turn, the model couples a suitable kinetics of... more
We consider a mathematical model for solidification of semicrystalline polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. In turn, the model couples a suitable kinetics of nonisothermal crystallization, taking into account both formation and growth of nuclei, with the thermal energy balance equation. We also present a model of secondary crystallization. The numerical approximation is performed by semiexplicit finite differences in time and finite elements in space. The fully discrete scheme amounts to solve, at any time step, a symmetric positive definite linear system preceded by an elementwise explicit computation. The computed numerical crystal structures match qualitatively the experimental ones.
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The evolution of a curvature dependent interface is approximated via a singularly perturbed parabolic double obstacle problem with small parameter ε>0. The velocity normal to the front is proportional to its mean curvature plus a... more
The evolution of a curvature dependent interface is approximated via a singularly perturbed parabolic double obstacle problem with small parameter ε>0. The velocity normal to the front is proportional to its mean curvature plus a forcing term. Optimal interface error estimates of order [Formula: see text] are derived for smooth evolutions, that is before singularities develop. Key ingredients are the construction of sub(super)-solutions containing several shape corrections dictated by formal asymptotics, and the use of a modified distance function.
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Research Interests: Mathematics, Materials Science, Mathematical Biology, Kinetics, Finite Element, and 15 moreMedicine, Biological Sciences, Numerical Simulation, Humans, Mathematical Sciences, Bone Cement, High Temperature, Mathematical Model, Finite Difference, Bone Cements, Medical Application, Boundary Value Problem, Optimality Condition, Heat Equation, and nonlinear equation
Research Interests: Engineering, Mathematics, Computational Physics, Computational Complexity, Finite element method, and 14 moreNumerical Simulation, Mathematical Sciences, Reaction-Diffusion Systems, Curvature, Physical sciences, Singularities, Mean Curvature Flow, Boundary Condition, Dynamic Mesh, Informing Science, Mean curvature, Minimal Surfaces, Driving force, and fattening
An asymptotic analysis is developed, which guarantees that the equation \u3b5a(x) 02u\u3b5/ 02t = \u3b5 divx(a(x)\u25bdxu\u3b5) - \u3c8(u\u3b5)/2\u3b5a(x) in Rn 7 (0, T), approximates a flow by mean curvature with an error of order... more
An asymptotic analysis is developed, which guarantees that the equation \u3b5a(x) 02u\u3b5/ 02t = \u3b5 divx(a(x)\u25bdxu\u3b5) - \u3c8(u\u3b5)/2\u3b5a(x) in Rn 7 (0, T), approximates a flow by mean curvature with an error of order O(\u3b52). The dependence on space of the relaxation parameter \u3b5a(x) is crucial for the stability and accuracy of the finite element approximations based on a local mesh refinement strategy. Several numerical experiments simulate the mean curvature motion of various surfaces and confirm the reliability of the asymptotic analysis
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An efficient implementation of finite element methods for free boundary parabolic problems in general two dimensional space domains is presented. The Stefan problem, the Hele‐Shaw problem and the porous medium equation are included.... more
An efficient implementation of finite element methods for free boundary parabolic problems in general two dimensional space domains is presented. The Stefan problem, the Hele‐Shaw problem and the porous medium equation are included. Backward differences or linearization techniques are used for the time discretization of the problem. The performances of these schemes are discussed with several numerical tests.
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Les effets d'hysteresis sont representes par des fonctionnelles de Volterra continues ou discontinues. Les relations constitutives de ce type sont couplees a des equations paraboliques et on obtient des resultats d'existence pour... more
Les effets d'hysteresis sont representes par des fonctionnelles de Volterra continues ou discontinues. Les relations constitutives de ce type sont couplees a des equations paraboliques et on obtient des resultats d'existence pour pour les formulations faibles correspondantes. On utilise une discretisation implicite du temps et des elements finis pour l'approximation numerique. Des tests numeriques montrent la convergence des solutions approchees
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... Stochastics, Mohrenstr. 39, 10117 Berlin, Germany AND CLAUDIO VERDI Dipartimento di Matematica, Universit`a di Milano, Via Saldini 50, 20133 Milano, Italy [Received on 17 September 2001; revised on 26 June 2002] A ...
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Quite precise asymptotic estimates, in terms of the relaxation parameter and the time step, are derived for travelling wave solutions to a Stefan problem with phase relaxation and a semidiscrete counterpart. These estimates quantify the... more
Quite precise asymptotic estimates, in terms of the relaxation parameter and the time step, are derived for travelling wave solutions to a Stefan problem with phase relaxation and a semidiscrete counterpart. These estimates quantify the regularizing effects of phase relaxation and time discretization that give rise to thin transition layers as opposed to sharp interfaces. Layer width estimates, pointwise error estimates, and asymptotic expressions for the profile of the relevant physical variables are proved. Applications to a related nonlinear Chernoff formula are also given.