Esaim-Probability and Statistics最新文献_第2页

Spectrum of the Dirichlet Laplacian in a thin cubic lattice 薄立方晶格中的狄利克雷拉普拉斯谱

4区数学

Esaim-Probability and Statistics Pub Date : 2023-10-05 DOI: 10.1051/m2an/2023082

Lucas Chesnel, Sergei A Nazarov

{"title":"Spectrum of the Dirichlet Laplacian in a thin cubic lattice","authors":"Lucas Chesnel, Sergei A Nazarov","doi":"10.1051/m2an/2023082","DOIUrl":"https://doi.org/10.1051/m2an/2023082","url":null,"abstract":"We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $epsll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+infty$ as $eps$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-delta/eps})$, $delta>0$, while the length of the next spectral segments is $O(eps)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^{Om}$ in the geometry $Om$ obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max-min arguments as well as a well-chosen Friedrichs inequality, we prove that $A^{Om}$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^{Om}$, that is no non trivial bounded solution at the threshold frequency for $A^{Om}$. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134946892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Stable approximation of Helmholtz solutions in the disk by evanescent plane waves 用消失的平面波稳定逼近圆盘中的亥姆霍兹解

4区数学

Esaim-Probability and Statistics Pub Date : 2023-10-04 DOI: 10.1051/m2an/2023081

Emile Parolin, Daan Huybrechs, Andrea Moiola

{"title":"Stable approximation of Helmholtz solutions in the disk by evanescent plane waves","authors":"Emile Parolin, Daan Huybrechs, Andrea Moiola","doi":"10.1051/m2an/2023081","DOIUrl":"https://doi.org/10.1051/m2an/2023081","url":null,"abstract":"Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz problems, aiming to achieve high accuracy with a small number of degrees of freedom. However, Trefftz methods lead to ill-conditioned linear systems, and it is often impossible to obtain the desired accuracy in floating-point arithmetic. In this paper we show that a judicious choice of plane waves can ensure high-accuracy solutions in a numerically stable way, in spite of having to solve such ill-conditioned systems. Numerical accuracy of plane wave methods is linked not only to the approximation space, but also to the size of the coefficients in the plane wave expansion. We show that the use of plane waves can lead to exponentially large coefficients, regardless of the orientations and the number of plane waves, and this causes numerical instability. We prove that all Helmholtz fields are continuous superposition of evanescent plane waves, i.e., plane waves with complex propagation vectors associated with exponential decay, and show that this leads to bounded representations. We provide a constructive scheme to select a set of real and complex-valued propagation vectors numerically. This results in an explicit selection of plane waves and an associated Trefftz method that achieves accuracy and stability. The theoretical analysis is provided for a two-dimensional domain with circular shape. However, the principles are general and we conclude the paper with a numerical experiment demonstrating practical applicability also for polygonal domains.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135548675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A new formulation of generalized gamma: Some results and applications 广义伽玛的一个新公式:一些结果和应用

4区数学

Esaim-Probability and Statistics Pub Date : 2023-10-03 DOI: 10.1051/ps/2023018

Cheng Peng

引用次数: 0

Convergence of a second-order scheme for non-local conservation laws 非局部守恒律二阶格式的收敛性

4区数学

Esaim-Probability and Statistics Pub Date : 2023-10-02 DOI: 10.1051/m2an/2023080

Sudarshan Kumar Kenettinkara, Nikhil Manoj, Veerappa Gowda G. D.

引用次数: 0

Exponential quasi-ergodicity for processes with discontinuous trajectories 不连续轨迹过程的指数拟遍历性

4区数学

Esaim-Probability and Statistics Pub Date : 2023-09-29 DOI: 10.1051/ps/2023016

Aurélien Velleret

引用次数: 1

A nonconforming immersed virtual element method for elliptic interface problems 求解椭圆界面问题的非协调浸入虚元法

4区数学

Esaim-Probability and Statistics Pub Date : 2023-09-15 DOI: 10.1051/m2an/2023078

Hyeokjoo Park, Do Young Kwak

引用次数: 0

Fully discrete pointwise smoothing error estimates for measure valued initial data 测量值初始数据的完全离散点平滑误差估计

4区数学

Esaim-Probability and Statistics Pub Date : 2023-09-06 DOI: 10.1051/m2an/2023076

Boris Vexler, Dmitriy Leykekhman, Jakob Wagner

引用次数: 0

VEM discretization allowing small edges for the reaction-convection-diffusion equation: source and spectral problems 允许小边缘反应-对流-扩散方程的VEM离散化:源和光谱问题

4区数学

Esaim-Probability and Statistics Pub Date : 2023-09-05 DOI: 10.1051/m2an/2023069

Felipe Lepe, Gonzalo Rivera

引用次数: 0

Pattern formation of the Cucker-Smale type kinetic models based on gradient flow 基于梯度流动的cucker - small型动力学模型的模式形成

4区数学

Esaim-Probability and Statistics Pub Date : 2023-09-01 DOI: 10.1051/m2an/2023079

Xinyu Wang, Xiaoping Xue

引用次数: 0

Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors 序批式反应器移动边界对流扩散反应模型的数值格式

4区数学

Esaim-Probability and Statistics Pub Date : 2023-09-01 DOI: 10.1051/m2an/2023068

Raimund Bürger, Julio Careaga, Stefan Diehl, Romel Pineda

{"title":"Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors","authors":"Raimund Bürger, Julio Careaga, Stefan Diehl, Romel Pineda","doi":"10.1051/m2an/2023068","DOIUrl":"https://doi.org/10.1051/m2an/2023068","url":null,"abstract":"Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136236636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0