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. © 2000 John Wiley & Sons, Inc.

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Abstract. 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.

Consider a closed surface in R n of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term. This geometric problem is rst approximated by a singularly perturbed parabolic double obstacle problem with small parameter " > 0. Conforming piecewise linear nite elements over a quasi-uniform and strongly acute mesh of size h are further used for space discretization, and combined with backward di erences for time discretization with uniform time-step. It is shown that the zero level set of the fully discrete solution converges past singularities to the true interface, provided ; h 2 o(" 3) and no fattening occurs. If the more stringent relations ; h 2 O(" 4) are enforced, then a linear rate of convergence O(") for interfaces is derived in the vicinity of regular points, namely those for which the underlying viscosity solution is nondegenerate. Singularities and their smearing e ect are also studied. The analysis is based on constructing discrete barriers via a parabolic projection, Lipschitz dependence of viscosity solutions with respect to perturbations of data, and discrete nondegeneracy. These issues are proven, along with quasi-optimality in 2D of the parabolic projection in L 1 with respect to both order and regularity requirements for functions in W 2;1 p .

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The implant of a femoral prosthesis is a critical process because of the relatively high temperature values reached at the bone/cement interface during the cementation of the infibulum. In fact, the cement is actually a polymer that polymerizes in situ generating heat. Moreover, the conversion of monomer into polymer is never 100%; this is dangerous because of the toxicity of the monomer. In this paper, we present a 3-D axisymmetric mathematical model capable of taking into account both the geometry of the implant and the chemical/physical properties of the cement. This model, together with its numerical simulation, thus represents a useful tool to set up the optimal conditions for the new materials developed in this orthopaedic field. The real complex geometry is assumed to be a bone/cement/metallic system having cylindrical symmetry, thus allowing the model to be reduced to two space variables. The cementation process is described by the Fourier heat equation coupled with a suitable polymerization kinetics. The numerical approximation is accomplished by semi-implicit finite differences in time and finite elements in space with numerical quadrature. The full discrete scheme amounts to solve linear positive definite symmetric systems preceded by an elementwise algebraic computation. We present various numerical simulations which confirm some critical aspects of this orthopaedic fixing technique such as thermal bone necrosis and the presence of unreacted residual monomer.

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