The propagation of the action potential in the heart chambers is accurately described bythe Bidomain model, which is commonly accepted and used in the specialistic literature.However, its mathematical structure of a degenerate parabolic system entails highcomputational costs in the numerical solution of the associated linear system. Domaindecomposition methods are a natural way to reduce computational costs, and OptimizedSchwarz Methods have proven in the recent years their effectiveness in accelerating theconvergence of such algorithms. The latter are based on interface matching conditions moreefficient than the classical Dirichlet or Neumann ones. In this paper we analyze anOptimized Schwarz approach for the numerical solution of the Bidomain problem. We assessthe convergence of the iterative method by means of Fourier analysis, and we investigatethe parameter optimization in the interface conditions. Numerical results in 2D and 3D aregiven to show the effectiveness of the method.

We propose a Lagrangian approach to deriving energy-preserving finite difference schemes for the Euler–Lagrange partial differential equations. Noether’s theorem states that the symmetry of time translation of Lagrangians yields the energy conservation law. We introduce a unique viewpoint on this theorem: “the symmetry of time translation of Lagrangians derives the Euler–Lagrange equation and the energy conservation law, simultaneously.” The proposed method is a combination of a discrete counter part of this statement and the discrete gradient method. It is also shown that the symmetry of space translation derives momentum-preserving schemes. Finally, we discuss the existence of discrete local conservation laws.

The aim of this paper is to analyze a formulation of the eddy current problem in terms of a time-primitive of the electric field in a bounded domain with input current intensities or voltage drops as source data. To this end, we introduce a Lagrange multiplier to impose the divergence-free condition in the dielectric domain. Thus, we obtain a time-dependent weak mixed formulation leading to a degenerate parabolic problem which we prove is well-posed. We propose a finite element method for space discretization based on Nédélec edge elements for the main variable and standard finite elements for the Lagrange multiplier, for which we obtain error estimates. Then, we introduce a backward Euler scheme for time discretization and prove error estimates for the fully discrete problem, too. Finally, the method is applied to solve a couple of test problems.

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.

The numerical approximation of parametric partial differential equations is acomputational challenge, in particular when the number of involved parameter is large.This paper considers a model class of second order, linear, parametric, elliptic PDEs on abounded domain D with diffusion coefficients depending on the parametersin an affine manner. For such models, it was shown in [9, 10] that under very weak assumptionson the diffusion coefficients, the entire family of solutions to such equations can besimultaneously approximated in the Hilbert space V = H01(D) by multivariate sparse polynomials in the parametervector y with a controlled number N of terms. Theconvergence rate in terms of N does not depend on the number ofparameters in V, which may be arbitrarily large or countably infinite,thereby breaking the curse of dimensionality. However, these approximation results do notdescribe the concrete construction of these polynomial expansions, and should thereforerather be viewed as benchmark for the convergence analysis of numerical methods. Thepresent paper presents an adaptive numerical algorithm for constructing a sequence ofsparse polynomials that is proved to converge toward the solution with the optimalbenchmark rate. Numerical experiments are presented in large parameter dimension, whichconfirm the effectiveness of the adaptive approach.

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectricstructure mounted on a perfectly conducting plate in three dimensions. Given thatuniqueness of solution holds, existence of solution follows from a well-known Fredholmframework for the variational formulation of the problem in a suitable Sobolev space. Inthis paper, we derive a Rellich identity for a solution to this variational problem undersuitable smoothness conditions on the material parameter. Under additional non-trappingassumptions on the material parameter, this identity allows us to establish uniqueness ofsolution for all positive wave numbers.

Discrete-velocity approximations represent a popular way for computing the Boltzmanncollision operator. The direct numerical evaluation of such methods involve a prohibitivecost, typically O(N2d + 1)where d is the dimension of the velocity space. In this paper, followingthe ideas introduced in [C. Mouhot and L. Pareschi, C. R. Acad. Sci. Paris Sér. IMath. 339 (2004) 71–76, C. Mouhot and L. Pareschi, Math.Comput. 75 (2006) 1833–1852], we derive fast summation techniquesfor the evaluation of discrete-velocity schemes which permits to reduce the computationalcost from O(N2d + 1) to O(N̅dNd log2N), N̅ ≪ N, with almost no loss of accuracy.

Space-time approximations of the FitzHugh–Nagumo system of coupled semi-linear parabolicPDEs are examined. The schemes under consideration are discontinuous in time butconforming in space and of arbitrary order. Stability estimates are presented in thenatural energy norms and at arbitrary times, under minimal regularity assumptions.Space-time error estimates of arbitrary order are derived, provided that the naturalparabolic regularity is present. Various physical parameters appearing in the model aretracked and numerical examples are presented.

We consider here the Interior Penalty Discontinuous Galerkin (IPDG) discretization of the wave equation. We show how to derive the optimal penalization parameter involved in this method in the case of regular meshes. Moreover, we provide necessary stability conditions of the global scheme when IPDG is coupled with the classical Leap–Frog scheme for the time discretization. Numerical experiments illustrate the fact that these conditions are also sufficient.

We analyze a numerical model for the Signorini unilateral contact, based on the mortarmethod, in the quadratic finite element context. The mortar frame enables one to usenon-matching grids and brings facilities in the mesh generation of different components ofa complex system. The convergence rates we state here are similar to those alreadyobtained for the Signorini problem when discretized on conforming meshes. The matching forthe unilateral contact driven by mortars preserves then the proper accuracy of thequadratic finite elements. This approach has already been used and proved to be reliablefor the unilateral contact problems even for large deformations. We provide however somenumerical examples to support the theoretical predictions.

We consider an asymptotic preserving numerical scheme initially proposed by F. Filbet andS. Jin [J. Comput. Phys. 229 (2010)] and G. Dimarco and L.Pareschi [SIAM J. Numer. Anal. 49 (2011) 2057–2077] in thecontext of nonlinear and stiff kinetic equations. Here, we propose a convergence analysisof such a scheme for the approximation of a system of transport equations with a nonlinearsource term, for which the asymptotic limit is given by a conservation law. We investigatethe convergence of the approximate solution (uεh, vεh) to a nonlinear relaxation system, whereε > 0 is a physical parameter andh represents the discretization parameter. Uniform convergence withrespect to ε and h is proved and error estimates arealso obtained. Finally, several numerical tests are performed to illustrate the accuracyand efficiency of such a scheme.

We consider the development and analysis of local discontinuous Galerkin methods forfractional diffusion problems in one space dimension, characterized by having fractionalderivatives, parameterized by β ∈[1, 2]. After demonstrating that aclassic approach fails to deliver optimal order of convergence, we introduce a modifiedlocal numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pureadvection (β = 1) and pure diffusion (β = 2). In the twoclassic limits, known schemes are recovered. We discuss stability and present an erroranalysis for the space semi-discretized scheme, which is supported through a fewexamples.

In this paper, we propose a numerical method to solve stochastic elliptic interfaceproblems with random interfaces. Shape calculus is first employed to derive theshape-Taylor expansion in the framework of the asymptotic perturbation approach. Given themean field and the two-point correlation function of the random interface, we can thusquantify the mean field and the variance of the random solution in terms of certain ordersof the perturbation amplitude by solving a deterministic elliptic interface problem andits tensorized counterpart with respect to the reference interface. Error estimates arederived for the interface-resolved finite element approximation in both, the physical andthe stochastic dimension. In particular, a fast finite difference scheme is proposed tocompute the variance of random solutions by using a low-rank approximation based on thepivoted Cholesky decomposition. Numerical experiments are presented to validate andquantify the method.

We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1and 1 − ε, where ε > 0 is a small real parameter,i.e. the waveguide is gently converging. The width of the junction zonefor the semi-strips is 1 + O(√ε). We will present a sufficient condition for the existence of a weaklycoupled bound state below π2, the lower bound of thecontinuous spectrum. This eigenvalue in the discrete spectrum is unique and itsasymptotics is constructed and justified whenε → 0+.

We consider the solution of second order elliptic PDEs in Rdwith inhomogeneous Dirichlet data by means of an h–adaptive FEM withfixed polynomial order p ∈ N. As model example serves the Poissonequation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneousDirichlet data are discretized by use of an H1 / 2–stableprojection, for instance, the L2–projection forp = 1 or the Scott–Zhang projection for general p ≥ 1.For error estimation, we use a residual error estimator which includes the Dirichlet dataoscillations. We prove that each H1 / 2–stable projectionyields convergence of the adaptive algorithm even with quasi–optimal convergence rate.Numerical experiments with the Scott–Zhang projection conclude the work.