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This paper proposes a statistical and probabilistic approach to compare and analyze the errors of two different approximation methods. We introduce the principle of numerical uncertainty in such a process, and we illustrate it by considering the discretization difference between two different approximation orders, e.g., first and second order Lagrangian finite element. Then, we derive a probabilistic approach to define and to qualify equivalent results. We illustrate our approach on a model problem on which we built the two above mentioned finite element approximations. We consider some variables as physical “predictors”, and we characterize how they influence the odds of the approximation methods to be locally “same order accurate”. [ABSTRACT FROM PUBLISHER]

We propose a new numerical method to compute the singular solution of the Maxwell equations in axisymmetric domains, as for example in non convex polygonal domains. As geometrical singularities are mainly related to the space dependent part of the model, we focus on the static field computation. We then introduce a new approach, that consists in decomposing the domain into two or more subdomains, and to derive an ad hoc variational formulation in each subdomain. The interface conditions are then imposed with a method deduced from a Nitsche method coupled with a specific “exchange” approach. An advantage of this domain decomposition method is that it does not require neither overlapping nor iteration process. Another advantage is that no particular mesh refinement is needed near the geometrical singularities. Numerical examples will be shown. [ABSTRACT FROM PUBLISHER]

This paper deals with the applications of data mining techniques in the evaluation of numerical solutions of Vlasov–Maxwell models. This is part of the topic of characterizing the model and approximation errors via learning techniques. We give two examples of application. The first one aims at comparing two Vlasov–Maxwell approximate models. In the second one, a scheme based on data mining techniques is proposed to characterize the errors between a P 1 and a P 2 finite element Particle-In-Cell approach. Beyond these examples, this original approach should operate in all cases where intricate numerical simulations like for the Vlasov–Maxwell equations take a central part. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a numerical method for handling boundary or transmission conditions in some linear partial differential equations. Depending on the nature of the conditions–essential, like the Dirichlet condition, or natural, like the Neumann condition–we derive a formulation based on a Nitsche approach together with an original exchange approach. We present our method first in a model problem, the Laplace problem, for which the Nitsche method was introduced to impose weakly essential boundary conditions. Then, we illustrate the method with two examples. First, the Maxwell equations, where the method shows its ability to handle singular solutions in non-convex domains. Then, an elasticity problem in a layered non-homogeneous domain, where the method proves to correctly take into account the transmission conditions at the interface between the layers and in the presence of cracks. [Copyright &y& Elsevier]

Abstract: This work deals with the crack problem simulation in dissimilar media. It proposes a new numerical method derived from a Nitsche approach for handling interface conditions in the elasticity equations. The Nitsche method, introduced to impose weakly essential boundary conditions in the scalar Laplace operator, has been then worked out more generally and transferred to continuity conditions. We propose here an extension of this method to the Navier–Lame equations. We derive a variational formulation that provides the solution in terms of displacements field in the case of a crack existence in a plate domain Ω, made of several different layers characterized by different material properties. We formulate the method for both the homogeneous and the dissimilar material domains and report some numerical experiments. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a method derived from a Nitsche approach for handling boundary and transmission conditions in some partial differential equations. Several years ago, the Nitsche method was introduced to impose weakly essential boundary conditions in the scalar Laplace operator. We propose here an extension to vector div −curl problems. Two examples of applications are presented. The first one is concerned with the Maxwell equations. This allows us to solve these equations, particularly in domains with reentrant corners, where the solution can be singular. The second example deals with the Navier-Lame equations. One can handle the case of a crack existence in a plate domain made of several different layers, characterized by different material properties. Numerical experiments are reported. [Copyright &y& Elsevier]

Abstract: This paper proposed a new approach based on data mining to evaluate the e_ciency of numerical asymptotic models. Indeed, data mining has proved to be an e_cient tool of analysis in several domains. In this work, we first derive an asymptotic paraxial approximation to model ultrarelativistic particles. Then, we use data mining methods that directly deal with numerical results of simulations, to understand what each order of the asymptotic expansion brings to the simulation results. This new approach o_ers the possibility to understand, on the numerical results themselves, the precision level of a numercial asymptotic model. [Copyright &y& Elsevier]

Abstract: We propose a new approach that consists in using data mining techniques for scientific computing. Indeed, data mining has proved to be efficient in other contexts which deal with huge data like in biology, medicine, marketing, advertising and communications. Our aim, here, is to deal with the important problem of the exploitation of the results produced by any numerical method. Indeed, more and more data are created today by numerical simulations. Thus, it seems necessary to look at efficient tools to analyze them. In this work, we focus our presentation to a test case dedicated to an asymptotic paraxial approximation to model ultrarelativistic particles. Our method directly deals with numerical results of simulations and try to understand what each order of the asymptotic expansion brings to the simulation results over what could be obtained by other lower-order or less accurate means. This new heuristic approach offers new potential applications to treat numerical solutions to mathematical models. [Copyright &y& Elsevier]

Abstract: We propose a novel approach that consists in using data mining techniques to perform a sensitivity analysis of approximate models. We give here the example of asymptotic solutions to Vlasov–Maxwell equations, obtained with a paraxial model for relativistic short beams. This new heuristic approach offers new potential applications to treat numerical solutions to mathematical models. [ABSTRACT FROM AUTHOR]

Abstract: We are concerned with the singular solution of the static Maxwell equation in a non-convex polygon. Thanks to a Hodge decomposition of the solution on a solenoidal and irrotational parts, one obtains an equivalent formulation to the static problem by solving two Laplace equations. Then a finite element formulation is derived, based on a Nitsche type method. This allows us to solve numerically the static Maxwell equation in domains with reentrant corners, where the solution can be singular. We formulate the method and report some numerical experiments. As a by product, this approach proves its ability to compute the dual singular functions of the Laplacian (see definition below). [Copyright &y& Elsevier]

Abstract: Solving the time-dependent Maxwell equations in an unbounded domain requires the introduction of artificial absorbing boundary conditions (ABCs) designed to minimize the amplitude of the parasitic waves reflected by the artificial frontier of the domain of computation. The construction of such ABCs needs to perform a rigorous mathematical and numerical analysis, in order to obtain a well-posed problem, from a mathematical point of view, and a stable algorithm, from a numerical point of view. In a previous study, Joly and Mercier (1989) have proposed a new second-order ABC for Maxwell’s equation in dimension 3, well adapted to a variational approach. In this paper, we present how to apply the second-order ABC proposed in in the framework of a finite element method. [Copyright &y& Elsevier]

Starting from classical absorbing boundary conditions, we propose a method for the separation of time-dependent scattered wave fields due to multiple sources or obstacles. In contrast to previous techniques, our method is local in space and time, deterministic, and avoids a priori assumptions on the frequency spectrum of the signal. Numerical examples in two space dimensions illustrate the usefulness of wave splitting for time-dependent scattering problems. [ABSTRACT FROM AUTHOR]