Claudio Verdi

Approximation of Degenerate Parabolic Problems Using Numerical Integration

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; nonstat...

A posteriori error estimates for variable time‐step discretizations of nonlinear evolution equations

We study the backward Euler method with variable time steps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the angle-bounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete, strongly nonlinear problems of parabolic type with degenerate or singular character.

A posteriori error estimation and adaptivity for degenerate parabolic problems

Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.

An adaptive finite element method for two-phase Stefan problems in two space dimensions. Part II.: implementation and numerical experiments

An adaptive local mesh refinement strategy for two-phase Stefan problems is discussed in light of its efficiency and computational complexity. Three local parameters are used to equidistribute interpolation errors in maximum norm for temperature and a fourth one, in the event of mushy regions, to equidistribute

Convergence Past Singularities for a Fully Discrete Approximation of Curvature-Drive Interfaces

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Numerical simulation of thermal bone necrosis during cementation of femoral prostheses

-interpolation errors for enthalpy within the mush. If certain quality mesh tests fail, then the current mesh is discarded and a new one completely regenerated by an efficient mesh generator, which in turn is briefly described. A typical triangulation is strongly graded to become very fine near computed interfaces and coarse away from them. Consecutive meshes are not compatible. The use of quadtree data structures is discussed as a means to reach a nearly optimal computational complexity in various tasks to be performed, mainly in generating a mesh and interpolating. Various implementation details are given so as to derive the computational complexity of each relevant subroutine. The approximation of both solutions and interfaces is drastically improved. The proposed method is robust in that it can handle the formation of cusps and mushy regions as well as the spontaneous appearance of phases. It is also superior in terms of computing time or a desired accuracy. Several numerical experiments illustrate these facts and provide quantitative information about each task complexity.

Error estimates for a semi-explicit numerical scheme for Stefan-type problems

Consider a closed surface in

SHARP ERROR ANALYSIS FOR CURVATURE DEPENDENT EVOLVING FRONTS

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Optimal error estimates for an approximation of degenerate parabolic problems

of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term. This geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter

The combined use of a nonlinear chernoff formula with a regularization procedure for two-phase stefan problems

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