This work considers stochastic Galerkin approximations of linear elliptic partial differential equations with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and infinity. A traditional numerical method for solving the resulting high-dimensional coupled system of partial differential equations (PDEs) is replaced by deep learning techniques. In order to achieve this, physics-informed neural networks (PINNs), which typically operate on the strong residual of the PDE and can therefore be applied in a wide range of settings, are considered. As a second approach, the Deep Ritz method, which is a neural network that minimizes the Ritz energy functional to find the weak solution, is employed. While the second approach only works in special cases, it overcomes the necessity of testing in variational problems while maintaining mathematical rigor and ensuring the existence of a unique solution. Furthermore, the residual is of a lower differentiation order, reducing the training cost considerably. The efficiency of the method is demonstrated on several model problems.
{"title":"Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs","authors":"Fabio Musco, Andrea Barth","doi":"arxiv-2409.08063","DOIUrl":"https://doi.org/arxiv-2409.08063","url":null,"abstract":"This work considers stochastic Galerkin approximations of linear elliptic\u0000partial differential equations with stochastic forcing terms and stochastic\u0000diffusion coefficients, that cannot be bounded uniformly away from zero and\u0000infinity. A traditional numerical method for solving the resulting\u0000high-dimensional coupled system of partial differential equations (PDEs) is\u0000replaced by deep learning techniques. In order to achieve this,\u0000physics-informed neural networks (PINNs), which typically operate on the strong\u0000residual of the PDE and can therefore be applied in a wide range of settings,\u0000are considered. As a second approach, the Deep Ritz method, which is a neural\u0000network that minimizes the Ritz energy functional to find the weak solution, is\u0000employed. While the second approach only works in special cases, it overcomes\u0000the necessity of testing in variational problems while maintaining mathematical\u0000rigor and ensuring the existence of a unique solution. Furthermore, the\u0000residual is of a lower differentiation order, reducing the training cost\u0000considerably. The efficiency of the method is demonstrated on several model\u0000problems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227624","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 paper deals with the numerical approximation of the Hilbert transform on the unit circle using Szeg"o and anti-Szeg"o quadrature formulas. These schemes exhibit maximum precision with oppositely signed errors and allow for improved accuracy through their averaged results. Their computation involves a free parameter associated with the corresponding para-orthogonal polynomials. Here, it is suitably chosen to construct a Szeg"o and anti-Szeg"o formula whose nodes are strategically distanced from the singularity of the Hilbert kernel. Numerical experiments demonstrate the accuracy of the proposed method.
{"title":"Approximation of the Hilbert Transform on the unit circle","authors":"Luisa Fermo, Valerio Loi","doi":"arxiv-2409.07810","DOIUrl":"https://doi.org/arxiv-2409.07810","url":null,"abstract":"The paper deals with the numerical approximation of the Hilbert transform on\u0000the unit circle using Szeg\"o and anti-Szeg\"o quadrature formulas. These\u0000schemes exhibit maximum precision with oppositely signed errors and allow for\u0000improved accuracy through their averaged results. Their computation involves a\u0000free parameter associated with the corresponding para-orthogonal polynomials.\u0000Here, it is suitably chosen to construct a Szeg\"o and anti-Szeg\"o formula\u0000whose nodes are strategically distanced from the singularity of the Hilbert\u0000kernel. Numerical experiments demonstrate the accuracy of the proposed method.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181924","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}
Singularly perturbed problems are known to have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods (FDMs), require a refined mesh to obtain stable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have been shown to successfully approximate solutions to differential equations from various fields, it is natural to examine their performance on singularly perturbed problems. The convection-diffusion equation is a representative example of such a class of problems, and we consider the use of PINNs to produce numerical solutions of this equation. We study two ways to use PINNS: as a method for correcting oscillatory discrete solutions obtained using FDMs, and as a method for modifying reduced solutions of unperturbed problems. For both methods, we also examine the use of input transformation to enhance accuracy, and we explain the behavior of input transformations analytically, with the help of neural tangent kernels.
{"title":"Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation","authors":"Jiajing Guan, Howard Elman","doi":"arxiv-2409.07671","DOIUrl":"https://doi.org/arxiv-2409.07671","url":null,"abstract":"Singularly perturbed problems are known to have solutions with steep boundary\u0000layers that are hard to resolve numerically. Traditional numerical methods,\u0000such as Finite Difference Methods (FDMs), require a refined mesh to obtain\u0000stable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have\u0000been shown to successfully approximate solutions to differential equations from\u0000various fields, it is natural to examine their performance on singularly\u0000perturbed problems. The convection-diffusion equation is a representative\u0000example of such a class of problems, and we consider the use of PINNs to\u0000produce numerical solutions of this equation. We study two ways to use PINNS:\u0000as a method for correcting oscillatory discrete solutions obtained using FDMs,\u0000and as a method for modifying reduced solutions of unperturbed problems. For\u0000both methods, we also examine the use of input transformation to enhance\u0000accuracy, and we explain the behavior of input transformations analytically,\u0000with the help of neural tangent kernels.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181926","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}
Francesco Della Santa, Antonio Mastropietro, Sandra Pieraccini, Francesco Vaccarino
The problem of multi-task regression over graph nodes has been recently approached through Graph-Instructed Neural Network (GINN), which is a promising architecture belonging to the subset of message-passing graph neural networks. In this work, we discuss the limitations of the Graph-Instructed (GI) layer, and we formalize a novel edge-wise GI (EWGI) layer. We discuss the advantages of the EWGI layer and we provide numerical evidence that EWGINNs perform better than GINNs over graph-structured input data with chaotic connectivity, like the ones inferred from the Erdos-R'enyi graph.
在这项工作中，我们讨论了图引导（GI）层的局限性，并正式提出了一种新颖的边缘引导 GI（EWGI）层。我们讨论了 EWGI 层的优势，并提供了数值证据，证明 EWGINN 在处理具有混沌连接性的图结构输入数据（如从 Erdos-R'enyi 图推断出的数据）时比 GINN 表现更好。
{"title":"Edge-Wise Graph-Instructed Neural Networks","authors":"Francesco Della Santa, Antonio Mastropietro, Sandra Pieraccini, Francesco Vaccarino","doi":"arxiv-2409.08023","DOIUrl":"https://doi.org/arxiv-2409.08023","url":null,"abstract":"The problem of multi-task regression over graph nodes has been recently\u0000approached through Graph-Instructed Neural Network (GINN), which is a promising\u0000architecture belonging to the subset of message-passing graph neural networks.\u0000In this work, we discuss the limitations of the Graph-Instructed (GI) layer,\u0000and we formalize a novel edge-wise GI (EWGI) layer. We discuss the advantages\u0000of the EWGI layer and we provide numerical evidence that EWGINNs perform better\u0000than GINNs over graph-structured input data with chaotic connectivity, like the\u0000ones inferred from the Erdos-R'enyi graph.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181927","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}
Charles L. Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang, Manas Rachh
We present a new complexification scheme based on the classical double layer potential for the solution of the Helmholtz equation with Dirichlet boundary conditions in compactly perturbed half-spaces in two and three dimensions. The kernel for the double layer potential is the normal derivative of the free-space Green's function, which has a well-known analytic continuation into the complex plane as a function of both target and source locations. Here, we prove that - when the incident data are analytic and satisfy a precise asymptotic estimate - the solution to the boundary integral equation itself admits an analytic continuation into specific regions of the complex plane, and satisfies a related asymptotic estimate (this class of data includes both plane waves and the field induced by point sources). We then show that, with a carefully chosen contour deformation, the oscillatory integrals are converted to exponentially decaying integrals, effectively reducing the infinite domain to a domain of finite size. Our scheme is different from existing methods that use complex coordinate transformations, such as perfectly matched layers, or absorbing regions, such as the gradual complexification of the governing wavenumber. More precisely, in our method, we are still solving a boundary integral equation, albeit on a truncated, complexified version of the original boundary. In other words, no volumetric/domain modifications are introduced. The scheme can be extended to other boundary conditions, to open wave guides and to layered media. We illustrate the performance of the scheme with two and three dimensional examples.
{"title":"Coordinate complexification for the Helmholtz equation with Dirichlet boundary conditions in a perturbed half-space","authors":"Charles L. Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang, Manas Rachh","doi":"arxiv-2409.06988","DOIUrl":"https://doi.org/arxiv-2409.06988","url":null,"abstract":"We present a new complexification scheme based on the classical double layer\u0000potential for the solution of the Helmholtz equation with Dirichlet boundary\u0000conditions in compactly perturbed half-spaces in two and three dimensions. The\u0000kernel for the double layer potential is the normal derivative of the\u0000free-space Green's function, which has a well-known analytic continuation into\u0000the complex plane as a function of both target and source locations. Here, we\u0000prove that - when the incident data are analytic and satisfy a precise\u0000asymptotic estimate - the solution to the boundary integral equation itself\u0000admits an analytic continuation into specific regions of the complex plane, and\u0000satisfies a related asymptotic estimate (this class of data includes both plane\u0000waves and the field induced by point sources). We then show that, with a\u0000carefully chosen contour deformation, the oscillatory integrals are converted\u0000to exponentially decaying integrals, effectively reducing the infinite domain\u0000to a domain of finite size. Our scheme is different from existing methods that\u0000use complex coordinate transformations, such as perfectly matched layers, or\u0000absorbing regions, such as the gradual complexification of the governing\u0000wavenumber. More precisely, in our method, we are still solving a boundary\u0000integral equation, albeit on a truncated, complexified version of the original\u0000boundary. In other words, no volumetric/domain modifications are introduced.\u0000The scheme can be extended to other boundary conditions, to open wave guides\u0000and to layered media. We illustrate the performance of the scheme with two and\u0000three dimensional examples.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181958","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 paper, we present a novel explicit second order scheme with one step for solving the forward backward stochastic differential equations, with the Crank-Nicolson method as a specific instance within our proposed framework. We first present a rigorous stability result, followed by precise error estimates that confirm the proposed novel scheme achieves second-order convergence. The theoretical results for the proposed methods are supported by numerical experiments.
{"title":"A novel second order scheme with one step for forward backward stochastic differential equations","authors":"Qiang Han, Shihao Lan, Quanxin Zhu","doi":"arxiv-2409.07118","DOIUrl":"https://doi.org/arxiv-2409.07118","url":null,"abstract":"In this paper, we present a novel explicit second order scheme with one step\u0000for solving the forward backward stochastic differential equations, with the\u0000Crank-Nicolson method as a specific instance within our proposed framework. We\u0000first present a rigorous stability result, followed by precise error estimates\u0000that confirm the proposed novel scheme achieves second-order convergence. The\u0000theoretical results for the proposed methods are supported by numerical\u0000experiments.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227625","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 outer inverse of tensors plays increasingly significant roles in computational mathematics, numerical analysis, and other generalized inverses of tensors. In this paper, we compute outer inverses with prescribed ranges and kernels of a given tensor through tensor QR decomposition and hyperpower iterative method under the M-product structure, which is a family of tensor-tensor products, generalization of the t-product and c-product, allows us to suit the physical interpretations across those different modes. We discuss a theoretical analysis of the nineteen-order convergence of the proposed tensor-based iterative method. Further, we design effective tensor-based algorithms for computing outer inverses using M-QR decomposition and hyperpower iterative method. The theoretical results are validated with numerical examples demonstrating the appropriateness of the proposed methods.
张量的外逆在计算数学、数值分析和其他张量的广义求逆中发挥着越来越重要的作用。在本文中，我们通过张量 QR 分解和超幂迭代法计算给定张量的具有规定范围和内核的外逆。M-product 结构是张量-张量乘积的一个族，是 t-product 和 c-product 的广义化，允许我们在这些不同模式之间进行物理解释。我们对所提出的基于张量的迭代法的十九阶收敛性进行了理论分析。此外，我们还设计了基于张量的有效算法，利用 M-QR 分解和超幂迭代法计算外倒数。我们用数值实例验证了理论结果，证明了所提方法的适用性。
{"title":"$M$-QR decomposition and hyperpower iterative methods for computing outer inverses of tensors","authors":"Ratikanta Behera, Krushnachandra Panigrahy, Jajati Keshari Sahoo, Yimin Wei","doi":"arxiv-2409.07007","DOIUrl":"https://doi.org/arxiv-2409.07007","url":null,"abstract":"The outer inverse of tensors plays increasingly significant roles in\u0000computational mathematics, numerical analysis, and other generalized inverses\u0000of tensors. In this paper, we compute outer inverses with prescribed ranges and\u0000kernels of a given tensor through tensor QR decomposition and hyperpower\u0000iterative method under the M-product structure, which is a family of\u0000tensor-tensor products, generalization of the t-product and c-product, allows\u0000us to suit the physical interpretations across those different modes. We\u0000discuss a theoretical analysis of the nineteen-order convergence of the\u0000proposed tensor-based iterative method. Further, we design effective\u0000tensor-based algorithms for computing outer inverses using M-QR decomposition\u0000and hyperpower iterative method. The theoretical results are validated with\u0000numerical examples demonstrating the appropriateness of the proposed methods.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181956","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}
Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick
We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations.
{"title":"Homogenisation for Maxwell and Friends","authors":"Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick","doi":"arxiv-2409.07084","DOIUrl":"https://doi.org/arxiv-2409.07084","url":null,"abstract":"We refine the understanding of continuous dependence on coefficients of\u0000solution operators under the nonlocal $H$-topology viz Schur topology in the\u0000setting of evolutionary equations in the sense of Picard. We show that certain\u0000components of the solution operators converge strongly. The weak convergence\u0000behaviour known from homogenisation problems for ordinary differential\u0000equations is recovered on the other solution operator components. The results\u0000are underpinned by a rich class of examples that, in turn, are also treated\u0000numerically, suggesting a certain sharpness of the theoretical findings.\u0000Analytic treatment of an example that proves this sharpness is provided too.\u0000Even though all the considered examples contain local coefficients, the main\u0000theorems and structural insights are of operator-theoretic nature and, thus,\u0000also applicable to nonlocal coefficients. The main advantage of the problem\u0000class considered is that they contain mixtures of type, potentially highly\u0000oscillating between different types of PDEs; a prototype can be found in\u0000Maxwell's equations highly oscillating between the classical equations and\u0000corresponding eddy current approximations.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181959","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 moist shallow water equations offer a promising route for advancing understanding of the coupling of physical parametrisations and dynamics in numerical atmospheric models, an issue known as 'physics-dynamics coupling'. Without moist physics, the traditional shallow water equations are a simplified form of the atmospheric equations of motion and so are computationally cheap, but retain many relevant dynamical features of the atmosphere. Introducing physics into the shallow water model in the form of moisture provides a tool to experiment with numerical techniques for physics-dynamics coupling in a simple dynamical model. In this paper, we compare some of the different moist shallow water models by writing them in a general formulation. The general formulation encompasses three existing forms of the moist shallow water equations and also a fourth, previously unexplored formulation. The equations are coupled to a three-state moist physics scheme that interacts with the resolved flow through source terms and produces two-way physics-dynamics feedback. We present a new compatible finite element discretisation of the equations and apply it to the different formulations of the moist shallow water equations in three test cases. The results show that the models capture generation of cloud and rain and physics-dynamics interactions, and demonstrate some differences between moist shallow water formulations and the implications of these different modelling choices.
{"title":"A compatible finite element discretisation for moist shallow water equations","authors":"Nell Hartney, Thomas M. Bendall, Jemma Shipton","doi":"arxiv-2409.07182","DOIUrl":"https://doi.org/arxiv-2409.07182","url":null,"abstract":"The moist shallow water equations offer a promising route for advancing\u0000understanding of the coupling of physical parametrisations and dynamics in\u0000numerical atmospheric models, an issue known as 'physics-dynamics coupling'.\u0000Without moist physics, the traditional shallow water equations are a simplified\u0000form of the atmospheric equations of motion and so are computationally cheap,\u0000but retain many relevant dynamical features of the atmosphere. Introducing\u0000physics into the shallow water model in the form of moisture provides a tool to\u0000experiment with numerical techniques for physics-dynamics coupling in a simple\u0000dynamical model. In this paper, we compare some of the different moist shallow\u0000water models by writing them in a general formulation. The general formulation\u0000encompasses three existing forms of the moist shallow water equations and also\u0000a fourth, previously unexplored formulation. The equations are coupled to a\u0000three-state moist physics scheme that interacts with the resolved flow through\u0000source terms and produces two-way physics-dynamics feedback. We present a new\u0000compatible finite element discretisation of the equations and apply it to the\u0000different formulations of the moist shallow water equations in three test\u0000cases. The results show that the models capture generation of cloud and rain\u0000and physics-dynamics interactions, and demonstrate some differences between\u0000moist shallow water formulations and the implications of these different\u0000modelling choices.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181930","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 develop and analyze a Reynolds-semi-robust and pressure-robust Hybrid High-Order (HHO) discretization of the incompressible Navier--Stokes equations. Reynolds-semi-robustness refers to the fact that, under suitable regularity assumptions, the right-hand side of the velocity error estimate does not depend on the inverse of the viscosity. This property is obtained here through a penalty term which involves a subtle projection of the convective term on a subgrid space constructed element by element. The estimated convergence order for the $L^infty(L^2)$- and $L^2(text{energy})$-norm of the velocity is $h^{k+frac12}$, which matches the best results for continuous and discontinuous Galerkin methods and corresponds to the one expected for HHO methods in convection-dominated regimes. Two-dimensional numerical results on a variety of polygonal meshes complete the exposition.
{"title":"A Reynolds-semi-robust and pressure robust Hybrid High-Order method for the time dependent incompressible Navier--Stokes equations on general meshes","authors":"Daniel Castanon Quiroz, Daniele A. Di Pietro","doi":"arxiv-2409.07037","DOIUrl":"https://doi.org/arxiv-2409.07037","url":null,"abstract":"In this work we develop and analyze a Reynolds-semi-robust and\u0000pressure-robust Hybrid High-Order (HHO) discretization of the incompressible\u0000Navier--Stokes equations. Reynolds-semi-robustness refers to the fact that,\u0000under suitable regularity assumptions, the right-hand side of the velocity\u0000error estimate does not depend on the inverse of the viscosity. This property\u0000is obtained here through a penalty term which involves a subtle projection of\u0000the convective term on a subgrid space constructed element by element. The\u0000estimated convergence order for the $L^infty(L^2)$- and\u0000$L^2(text{energy})$-norm of the velocity is $h^{k+frac12}$, which matches the\u0000best results for continuous and discontinuous Galerkin methods and corresponds\u0000to the one expected for HHO methods in convection-dominated regimes.\u0000Two-dimensional numerical results on a variety of polygonal meshes complete the\u0000exposition.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181955","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：