In this paper we present two versions of the central localdiscontinuous Galerkin (LDG) method on overlapping cellsfor solving diffusion equations, and provide theirstability analysis and error estimates for the linear heat equation.A comparisonbetween the traditional LDG method ona single mesh and the two versions of the central LDGmethod on overlapping cells is also made.Numerical experiments are provided to validate the quantitativeconclusions from the analysis and to support conclusions forgeneral polynomial degrees.

We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem.It has been developed following two main ideas.On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes.On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique.We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order.With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions.Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

The Bidomain model is nowadays one of the most accurate mathematical descriptions of the action potential propagation in the heart. However, its numerical approximation is in general fairly expensive as a consequence of the mathematical features of this system. For this reason, a simplification of this model, called Monodomain problem is quite often adopted in order to reduce computational costs. Reliability of this model is however questionable, in particular in the presence of applied currents and in the regions where the upstroke or the late recovery of the action potential is occurring. In this paper we investigate a domain decomposition approach for this problem, where the entire computational domain is suitably split and the two models are solved in different subdomains. Since the mathematical features of the two problems are rather different, the heterogeneous coupling is non trivial. Here we investigate appropriate interface matching conditions for the coupling on non overlapping domains. Moreover, we pursue an Optimized Schwarz approach for the numerical solution of the heterogeneous problem. Convergence of the iterative method is analyzed by means of a Fourier analysis. We investigate the parameters to be selected in the matching radiation-type conditions to accelerate the convergence. Numerical results both in two and three dimensions illustrate the effectiveness of the coupling strategy.

By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec's edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proofis new, instead, and shows how families of spurious-freeelements can be systematically constructed. The tools here developed are used to define a new family of spurious-freeedge elements which, in some sense, are complementary to those defined in 1986 by Nedelec.

This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of ${\mathbb R}^d$ . This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.

We construct finite volume schemes, on unstructured and irregular grids andin any space dimension, for non-linear elliptic equations of the p-Laplacian kind: -div(|∇u|p-2 ∇u) = ƒ(with 1 < p < ∞). We prove the existence and uniqueness of the approximate solutions,as well as their strong convergence towards the solution of the PDE.The outcome of some numerical tests are also provided.

In this article, we derive a complete mathematical analysis of acoupled 1D-2D model for 2D wave propagation in media including thin slots. Our error estimates are illustrated by numerical results.

We derive a constitutive law for the myocardium from the description of both the geometrical arrangement ofcardiomyocytes and their individual mechanical behaviour. We model a set of cardiomyocytes by a quasiperiodic discretelattice of elastic bars interacting by means of moments. We work in a large displacement framework and we use a discretehomogenization technique. The macroscopic constitutive law is obtained through the resolution of anonlinear self-equilibrum system of the discrete lattice reference cell.

We present an improved, near-optimal hp error estimate for a non-conforming finite element method, called the mortar method (M0). We also present a new hp mortaring technique, called the mortar method (MP), and derive h, p and hp error estimates for it, in the presence of quasiuniform and non-quasiuniform meshes. Our theoretical results, augmented by the computational evidence we present, show that like (M0), (MP) is also a viable mortaring technique for the hp method.

Tuning the alternating Schwarz method to the exterior problems is the subject of this paper. We present the original algorithm and we propose a modification of it, so that the solution of the subproblem involving the condition at infinity has an explicit integral representation formulas while the solution of the other subproblem, set in a bounded domain, is approximated by classical variational methods. We investigate many of the advantages of the new Schwarz approach: a geometrical convergence rate, an easy implementation, a substantial economy in computational costs and a satisfactory accuracy in the numerical results as well as their agreement with the theoretical statements.

We address the issue of parameter variations in POD approximations of time-dependentproblems, without any specific restriction on the form of parameter dependence.Considering a parabolic model problem, we propose a POD construction strategy allowing usto obtain some a priori error estimates controlled by the POD remainder –in the construction procedure – and some parameter-wise interpolation errors for the modelsolutions. We provide a thorough numerical assessment of this strategy with theFitzHugh − Nagumo 1D model. Finally, we give detailed illustrations of the approach in twoparameter estimation applications, the first in a variational estimation framework withthe FitzHugh − Nagumo model, and the second with a beating heart mechanical model forwhich we employ a sequential estimation method to characterize model parameters using realimage data in a clinical case.

This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a desired temperature can be optimally achieved. As there are finitely many control parameters but the state constraint has to be satisfied in an infinite number of points, the problem belongs to a class of semi-infinite programming problems. We present a rigorous analysis of the optimization problem and a numerical strategy based on our theoretical result.


We consider a special configuration of vorticity that consists of a pair ofexternally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary weak solution. Preliminary simulations presented here suggest this isindeed the case. We establish a convergence theorem for the vortex blob method that applies to this problem. This theorem and the preliminary calculations we carried out support the existence of two distinct weak solutions with the same initial data.


Mathematical models for option pricing often result in partial differential equations.Recent enhancements are models driven by Lévy processes, which lead to a partialdifferential equation with an additional integral term. In the context of modelcalibration, these partial integro differential equations need to be solved quitefrequently. To reduce the computational cost the implementation of a reduced order modelhas shown to be very successful numerically. In this paper we give a priorierror estimates for the use of the proper orthogonal decomposition technique inthe context of option pricing models.

We consider the following problem of error estimation for the optimal control ofnonlinear parabolic partial differential equations: let an arbitrary admissible controlfunction be given. How far is it from the next locally optimal control? Under naturalassumptions including a second-order sufficient optimality condition for the (unknown)locally optimal control, we estimate the distance between the two controls. To do this, weneed some information on the lowest eigenvalue of the reduced Hessian. We apply thistechnique to a model reduced optimal control problem obtained by proper orthogonaldecomposition (POD). The distance between a local solution of the reduced problem to alocal solution of the original problem is estimated.

We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatly improves accuracy in simulations. Standard finite element schemes for NS-α suffer from two major sources of error if their solutions are considered approximations to true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure error on the velocity error that arises from the (necessary) use of the rotational form nonlinearity. The proposed scheme “fixes” these two numerical issues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We prove the scheme is stable, optimally convergent, and the effect of the pressure error on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate the effectiveness of the method.

We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed.In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of thediscussion, we give a recipe that ensures that the number of extrema within the discretized version of the initial data cannot increase by the application of the scheme. The usefulness of the recipe is confirmed by numerical tests.

This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.

We present a new methodology for the numerical resolution of the hydrodynamicsof incompressible viscid newtonian fluids. It is based on the Navier-Stokesequations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurationstypical to the motion of biological structures in viscous fluids.Although the method is applicable to three dimensions, we address herein detail only the two dimensional case. We provide numerical data forsome test cases to which we apply the computational scheme.

We consider a variational formulation of the three-dimensional Navier–Stokes equations with mixed boundary conditions and prove that the variational problem admits a solution provided that the domain satisfies a suitable regularity assumption. Next, we propose a finite element discretization relying on the Galerkin method and establish a priori and a posteriori error estimates.