Eligio Colmenares, G. Gatica, S. Moraga, R. Ruiz-Baier
A new fully-mixed formulation is advanced for the stationary Oberbeck–Boussinesq problem when viscosity depends on both temperature and concentration of a solute. Following recent ideas in the context of mixed methods for Boussinesq and Navier–Stokes systems, the velocity gradient and the Bernoulli stress tensor are taken as additional field variables in the momentum and mass equilibrium equations. Similarly, the gradients of temperature and concentration together with a Bernoulli vector are considered as unknowns in the heat and mass transfer equations. Consequently, a dual-mixed approach with Dirichlet data is defined in each sub-system, and the well-known Banach and Brouwer theorems are combined with Babuška–Brezzi’s theory in each independent set of equations, yielding the solvability of the continuous and discrete schemes. We show that our development also applies to the case where the equations of thermal energy and solute transport are coupled through cross-diffusion. Appropriate finite element subspaces are specified, and optimal a priori error estimates are derived. Furthermore, a reliable and efficient residual-based a posteriori error estimator is proposed. Several numerical examples illustrate the performance of the fully-mixed scheme and of the adaptive refinement algorithm driven by the error estimator. 2020 Mathematics Subject Classification. 65N30, 65N12, 65N15, 35Q79, 80A20, 76D05, 76R10.
{"title":"A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system","authors":"Eligio Colmenares, G. Gatica, S. Moraga, R. Ruiz-Baier","doi":"10.5802/smai-jcm.64","DOIUrl":"https://doi.org/10.5802/smai-jcm.64","url":null,"abstract":"A new fully-mixed formulation is advanced for the stationary Oberbeck–Boussinesq problem when viscosity depends on both temperature and concentration of a solute. Following recent ideas in the context of mixed methods for Boussinesq and Navier–Stokes systems, the velocity gradient and the Bernoulli stress tensor are taken as additional field variables in the momentum and mass equilibrium equations. Similarly, the gradients of temperature and concentration together with a Bernoulli vector are considered as unknowns in the heat and mass transfer equations. Consequently, a dual-mixed approach with Dirichlet data is defined in each sub-system, and the well-known Banach and Brouwer theorems are combined with Babuška–Brezzi’s theory in each independent set of equations, yielding the solvability of the continuous and discrete schemes. We show that our development also applies to the case where the equations of thermal energy and solute transport are coupled through cross-diffusion. Appropriate finite element subspaces are specified, and optimal a priori error estimates are derived. Furthermore, a reliable and efficient residual-based a posteriori error estimator is proposed. Several numerical examples illustrate the performance of the fully-mixed scheme and of the adaptive refinement algorithm driven by the error estimator. 2020 Mathematics Subject Classification. 65N30, 65N12, 65N15, 35Q79, 80A20, 76D05, 76R10.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128324194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Augmented Lagrangian preconditioners have successfully yielded Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes equations, but only for specific discretizations. The discretizations for which these preconditioners have been designed possess error estimates which depend on the Reynolds number, with the discretization error deteriorating as the Reynolds number is increased. In this paper we present an augmented Lagrangian preconditioner for the Scott-Vogelius discretization on barycentrically-refined meshes. This achieves both Reynolds-robust performance and Reynolds-robust error estimates. A key consideration is the design of a suitable space decomposition that captures the kernel of the grad-div term added to control the Schur complement; the same barycentric refinement that guarantees inf-sup stability also provides a local decomposition of the kernel of the divergence. The robustness of the scheme is confirmed by numerical experiments in two and three dimensions.
{"title":"A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations","authors":"P. Farrell, L. Mitchell, L. R. Scott, F. Wechsung","doi":"10.5802/smai-jcm.72","DOIUrl":"https://doi.org/10.5802/smai-jcm.72","url":null,"abstract":"Augmented Lagrangian preconditioners have successfully yielded Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes equations, but only for specific discretizations. The discretizations for which these preconditioners have been designed possess error estimates which depend on the Reynolds number, with the discretization error deteriorating as the Reynolds number is increased. In this paper we present an augmented Lagrangian preconditioner for the Scott-Vogelius discretization on barycentrically-refined meshes. This achieves both Reynolds-robust performance and Reynolds-robust error estimates. A key consideration is the design of a suitable space decomposition that captures the kernel of the grad-div term added to control the Schur complement; the same barycentric refinement that guarantees inf-sup stability also provides a local decomposition of the kernel of the divergence. The robustness of the scheme is confirmed by numerical experiments in two and three dimensions.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133755247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce some IMEX schemes (implicit-explicit schemes with an implicit term being linear) for approximating elastodynamic contact problems when the contact condition is taken into account with a Nitsche method. We develop a theoretical and numerical study of the properties of the schemes, especially in terms of stability, provide some numerical comparisons with standard explicit and implicit scheme and propose some improvements to obtain a more reliable approximation of motion for large time steps. We also show how selective mass scaling techniques can be interpreted as IMEX schemes.
{"title":"Stable IMEX schemes for a Nitsche-based approximation of elastodynamic contact problems. Selective mass scaling interpretation","authors":"É. Bretin, Y. Renard","doi":"10.5802/SMAI-JCM.65","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.65","url":null,"abstract":"We introduce some IMEX schemes (implicit-explicit schemes with an implicit term being linear) for approximating elastodynamic contact problems when the contact condition is taken into account with a Nitsche method. We develop a theoretical and numerical study of the properties of the schemes, especially in terms of stability, provide some numerical comparisons with standard explicit and implicit scheme and propose some improvements to obtain a more reliable approximation of motion for large time steps. We also show how selective mass scaling techniques can be interpreted as IMEX schemes.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129637956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Bayesian formulation of inverse problems is attractive for three primary reasons: it provides a clear modelling framework; means for uncertainty quantification; and it allows for principled learning of hyperparameters. The posterior distribution may be explored by sampling methods, but for many problems it is computationally infeasible to do so. In this situation maximum a posteriori (MAP) estimators are often sought. Whilst these are relatively cheap to compute, and have an attractive variational formulation, a key drawback is their lack of invariance under change of parameterization. This is a particularly significant issue when hierarchical priors are employed to learn hyperparameters. In this paper we study the effect of the choice of parameterization on MAP estimators when a conditionally Gaussian hierarchical prior distribution is employed. Specifically we consider the centred parameterization, the natural parameterization in which the unknown state is solved for directly, and the noncentred parameterization, which works with a whitened Gaussian as the unknown state variable, and arises when considering dimension-robust MCMC algorithms; MAP estimation is well-defined in the nonparametric setting only for the noncentred parameterization. However, we show that MAP estimates based on the noncentred parameterization are not consistent as estimators of hyperparameters; conversely, we show that limits of finite-dimensional centred MAP estimators are consistent as the dimension tends to infinity. We also consider empirical Bayesian hyperparameter estimation, show consistency of these estimates, and demonstrate that they are more robust with respect to noise than centred MAP estimates. An underpinning concept throughout is that hyperparameters may only be recovered up to measure equivalence, a well-known phenomenon in the context of the Ornstein-Uhlenbeck process.
{"title":"Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency","authors":"Matthew M. Dunlop, T. Helin, A. Stuart","doi":"10.5802/smai-jcm.62","DOIUrl":"https://doi.org/10.5802/smai-jcm.62","url":null,"abstract":"The Bayesian formulation of inverse problems is attractive for three primary reasons: it provides a clear modelling framework; means for uncertainty quantification; and it allows for principled learning of hyperparameters. The posterior distribution may be explored by sampling methods, but for many problems it is computationally infeasible to do so. In this situation maximum a posteriori (MAP) estimators are often sought. Whilst these are relatively cheap to compute, and have an attractive variational formulation, a key drawback is their lack of invariance under change of parameterization. This is a particularly significant issue when hierarchical priors are employed to learn hyperparameters. In this paper we study the effect of the choice of parameterization on MAP estimators when a conditionally Gaussian hierarchical prior distribution is employed. Specifically we consider the centred parameterization, the natural parameterization in which the unknown state is solved for directly, and the noncentred parameterization, which works with a whitened Gaussian as the unknown state variable, and arises when considering dimension-robust MCMC algorithms; MAP estimation is well-defined in the nonparametric setting only for the noncentred parameterization. However, we show that MAP estimates based on the noncentred parameterization are not consistent as estimators of hyperparameters; conversely, we show that limits of finite-dimensional centred MAP estimators are consistent as the dimension tends to infinity. We also consider empirical Bayesian hyperparameter estimation, show consistency of these estimates, and demonstrate that they are more robust with respect to noise than centred MAP estimates. An underpinning concept throughout is that hyperparameters may only be recovered up to measure equivalence, a well-known phenomenon in the context of the Ornstein-Uhlenbeck process.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114823973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Second order macroscopic traffic flow models are able to reproduce the so-called capacity drop effect, i.e., the phenomenon that the outflow of a congested region is substantially lower than the maximum achievable flow. Within this work, we propose a first order model for a junction with ramp buffer that is solely modified at the intersection so that the capacity drop is captured. Theoretical investigations motivate the new choice of coupling conditions and illustrate the difference to purely first and second order models. The numerical example considering the optimal control of the onramp merging into a main road highlights that the combined model generates similar results as the second order model. AMS subject classifications: 65M08, 90C30
{"title":"A combined first and second order model for a junction with ramp buffer","authors":"J. Weißen, O. Kolb, Simone Gottlich","doi":"10.5802/smai-jcm.90","DOIUrl":"https://doi.org/10.5802/smai-jcm.90","url":null,"abstract":"Second order macroscopic traffic flow models are able to reproduce the so-called capacity drop effect, i.e., the phenomenon that the outflow of a congested region is substantially lower than the maximum achievable flow. Within this work, we propose a first order model for a junction with ramp buffer that is solely modified at the intersection so that the capacity drop is captured. Theoretical investigations motivate the new choice of coupling conditions and illustrate the difference to purely first and second order models. The numerical example considering the optimal control of the onramp merging into a main road highlights that the combined model generates similar results as the second order model. AMS subject classifications: 65M08, 90C30","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126991327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we analyse the convergence behaviour and scalability properties of the one-level Parallel Schwarz method (PSM) for domain decomposition problems in which the boundaries of many subdomains lie in the interior of the global domain. Such problems arise, for instance, in solvation models in computational chemistry. Existing results on the scalability of the one-level PSM are limited to situations where each subdomain has access to the external boundary, and at most only two subdomains have a common overlap. We develop a systematic framework that allows us to bound the norm of the Schwarz iteration operator for domain decomposition problems in which subdomains may be completely embedded in the interior of the global domain and an arbitrary number of subdomains may have a common overlap.
{"title":"On the Scalability of the Schwarz Method","authors":"G. Ciaramella, Muhammad Hassan, B. Stamm","doi":"10.5802/smai-jcm.61","DOIUrl":"https://doi.org/10.5802/smai-jcm.61","url":null,"abstract":"In this article, we analyse the convergence behaviour and scalability properties of the one-level Parallel Schwarz method (PSM) for domain decomposition problems in which the boundaries of many subdomains lie in the interior of the global domain. Such problems arise, for instance, in solvation models in computational chemistry. Existing results on the scalability of the one-level PSM are limited to situations where each subdomain has access to the external boundary, and at most only two subdomains have a common overlap. We develop a systematic framework that allows us to bound the norm of the Schwarz iteration operator for domain decomposition problems in which subdomains may be completely embedded in the interior of the global domain and an arbitrary number of subdomains may have a common overlap.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125104985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-$q$-monotone approximation with $qge 3$ is completely different from comonotone and coconvex cases. Additionally, we show that, for each function $f$ from $Delta^{(1)}$, the set of all monotone functions on $[-1,1]$, and every $alpha>0$, we have [ limsup_{ntoinfty} inf_{P_ninmathbb P_ncapDelta^{(1)}} left| frac{n^alpha(f-P_n)}{varphi^alpha} right| le c(alpha) limsup_{ntoinfty} inf_{P_ninmathbb P_n} left| frac{n^alpha(f-P_n)}{varphi^alpha} right| ] where $mathbb P_n$ denotes the set of algebraic polynomials of degree $
{"title":"Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update","authors":"K. Kopotun, D. Leviatan, I. Shevchuk","doi":"10.5802/smai-jcm.54","DOIUrl":"https://doi.org/10.5802/smai-jcm.54","url":null,"abstract":"It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. \u0000The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-$q$-monotone approximation with $qge 3$ is completely different from comonotone and coconvex cases. \u0000Additionally, we show that, for each function $f$ from $Delta^{(1)}$, the set of all monotone functions on $[-1,1]$, and every $alpha>0$, we have \u0000[ limsup_{ntoinfty} inf_{P_ninmathbb P_ncapDelta^{(1)}} left| frac{n^alpha(f-P_n)}{varphi^alpha} right| le c(alpha) limsup_{ntoinfty} inf_{P_ninmathbb P_n} left| frac{n^alpha(f-P_n)}{varphi^alpha} right| ] \u0000where $mathbb P_n$ denotes the set of algebraic polynomials of degree $","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127847267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we focus on the development of the use of Periodic Boundary Conditions (PBC) with sources at oblique incidence in a Discontinuous Galerkin Time Domain (DGTD) framework. Whereas in the context of the Finite Difference Time Domain (FDTD) methods, an abundant literature can be found, for DGTD, the amount of contributions reporting on such methods is remarkably low. In this paper, we supplement the existing references using the field transform technique with an analysis of the continuous system using the method of characteristics and provide an energy estimate. Furthermore, we also study the discrete stability of the resulting DGTD scheme. Additional details about sources, observables (reflectance, transmittance and diffraction efficiency), and the use of Complex Frequency-Shifted Perfectly-Matched Layers (CFS-PMLs) in this framework are also given. After numerical validations, two realistic test-cases are considered in the context of nanophotonics with the Diogenes DGTD solver (http://diogenes.inria.fr).
{"title":"Simulating 3D periodic structures at oblique incidences with discontinuous Galerkin time-domain methods: theoretical and practical considerations","authors":"J. Viquerat, N. Schmitt, C. Scheid","doi":"10.5802/SMAI-JCM.45","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.45","url":null,"abstract":"In this work, we focus on the development of the use of Periodic Boundary Conditions (PBC) with sources at oblique incidence in a Discontinuous Galerkin Time Domain (DGTD) framework. Whereas in the context of the Finite Difference Time Domain (FDTD) methods, an abundant literature can be found, for DGTD, the amount of contributions reporting on such methods is remarkably low. In this paper, we supplement the existing references using the field transform technique with an analysis of the continuous system using the method of characteristics and provide an energy estimate. Furthermore, we also study the discrete stability of the resulting DGTD scheme. Additional details about sources, observables (reflectance, transmittance and diffraction efficiency), and the use of Complex Frequency-Shifted Perfectly-Matched Layers (CFS-PMLs) in this framework are also given. After numerical validations, two realistic test-cases are considered in the context of nanophotonics with the Diogenes DGTD solver (http://diogenes.inria.fr).","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130852339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main goal of this paper is to provide a brief survey of recent results which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to the discrepancy theory. We discuss this connection in detail and provide a general view of this connection. It was established recently that the new concept of {it fixed volume discrepancy} is very useful in proving the upper bounds for the dispersion. Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.
{"title":"Connections between numerical integration, discrepancy, dispersion, and universal discretization","authors":"V. Temlyakov","doi":"10.5802/smai-jcm.58","DOIUrl":"https://doi.org/10.5802/smai-jcm.58","url":null,"abstract":"The main goal of this paper is to provide a brief survey of recent results which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to the discrepancy theory. We discuss this connection in detail and provide a general view of this connection. It was established recently that the new concept of {it fixed volume discrepancy} is very useful in proving the upper bounds for the dispersion. Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115978019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
1 Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi 55, I-20153, Milano, Italy and IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy E-mail address: lourenco.beirao@unimib.it 2 IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy E-mail address: brezzi@imati.cnr.it 3 Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, I-27100 Pavia, Italy and IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy E-mail address: marini@imati.cnr.it 4 Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi 55, I-20153, Milano, Italy and IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy E-mail address: alessandro.russo@unimib.it.
1、大学应用数学和米兰Bicocca、Cozzi 55 I-20153、米兰、意大利和IMATI-CNR、意大利帕维亚I-27100方面1、电子邮件为:lourenco beirao@unimib。it 2 IMATI-CNR,意大利帕维亚I-27100方面1、电子邮件为:数学brezzi@imati .中央人民广播电台3系、帕维亚大学、中法大学5、I-27100 Pavia, Italy and IMATI-CNR,中法1、意大利帕维亚I-27100,电子邮件为:marini@imati.cnr.it 4数学与应用系，米兰大学Bicocca, Via Cozzi 55, I-20153，米兰，意大利和意大利- cnr, Via Ferrata 1, I-27100 Pavia，意大利电子邮件address: alessandro.russo@unimib。
{"title":"Virtual Element approximations of the Vector Potential Formulation of Magnetostatic problems","authors":"L. B. D. Veiga, F. Brezzi, L. D. Marini, A. Russo","doi":"10.5802/SMAI-JCM.40","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.40","url":null,"abstract":"1 Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi 55, I-20153, Milano, Italy and IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy E-mail address: lourenco.beirao@unimib.it 2 IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy E-mail address: brezzi@imati.cnr.it 3 Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, I-27100 Pavia, Italy and IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy E-mail address: marini@imati.cnr.it 4 Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi 55, I-20153, Milano, Italy and IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy E-mail address: alessandro.russo@unimib.it.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115256124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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