This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov–Stokes system with periodic boundary condition. The method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation. The proposed method is both mass and momentum conservative. Since it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes operator become non-coercive and indefinite, and under the smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes projection to deal with the Stokes part and, with the help of a special projection, to tackle the Vlasov part. Finally, numerical experiments based on the dG method combined with a splitting algorithm are performed.
{"title":"Discontinuous Galerkin Methods for the Vlasov–Stokes System","authors":"Harsha Hutridurga, Krishan Kumar, Amiya K. Pani","doi":"10.1515/cmam-2023-0243","DOIUrl":"https://doi.org/10.1515/cmam-2023-0243","url":null,"abstract":"This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov–Stokes system with periodic boundary condition. The method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation. The proposed method is both mass and momentum conservative. Since it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes operator become non-coercive and indefinite, and under the smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes projection to deal with the Stokes part and, with the help of a special projection, to tackle the Vlasov part. Finally, numerical experiments based on the dG method combined with a splitting algorithm are performed.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"5 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646424","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}
We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system. We prove that the 𝑊-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains. Numerical results are shown for both 𝑊-cycle and 𝑉-cycle algorithms.
{"title":"Robust Multigrid Methods for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem","authors":"Sijing Liu","doi":"10.1515/cmam-2023-0132","DOIUrl":"https://doi.org/10.1515/cmam-2023-0132","url":null,"abstract":"We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system. We prove that the 𝑊-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains. Numerical results are shown for both 𝑊-cycle and 𝑉-cycle algorithms.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"32 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646768","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}
We compare different machine learning estimators and present details about their implementation in Python. The computational studies are conducted for classification as well as regression problems. Moreover, as one of the founding problems of machine learning, we present the specific classification task of handwritten digit recognition. In this connection, we discuss the mathematical formulation and of course the implementation details of this problem. All corresponding Python code is fully provided on request and can be downloaded from the author’s GitHub page https://github.com/Fab1Fatal.
{"title":"Machine Learning Estimators: Implementation and Comparison in Python","authors":"Fabian Merle","doi":"10.1515/cmam-2023-0198","DOIUrl":"https://doi.org/10.1515/cmam-2023-0198","url":null,"abstract":"We compare different <jats:italic>machine learning</jats:italic> estimators and present details about their implementation in Python. The computational studies are conducted for <jats:italic>classification</jats:italic> as well as <jats:italic>regression</jats:italic> problems. Moreover, as one of the founding problems of machine learning, we present the specific <jats:italic>classification</jats:italic> task of <jats:italic>handwritten digit recognition</jats:italic>. In this connection, we discuss the mathematical formulation and of course the implementation details of this problem. All corresponding Python code is fully provided on request and can be downloaded from the author’s GitHub page <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"uri\" xlink:href=\"https://github.com/Fab1Fatal\">https://github.com/Fab1Fatal</jats:ext-link>.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"7 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557710","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}
This paper deals with the efficient implementation of the finite element method with continuous piecewise linear functions (P1-FEM) in Rdmathbb{R}^{d} (d∈Ndinmathbb{N}). Although at present there does not seem to be a very high practical demand for finite element methods that use higher-dimensional simplicial partitions, there are some advantages in studying the efficient implementation of the method independent of the dimension. For instance, it provides additional insights into necessary data structures and the complexity of implementations. Throughout, the focus is on an efficient realization using Matlab built-in functions and vectorization. The fast and vectorized Matlab function can be easily implemented in many other vector languages and is provided in Julia, too. The complete implementation of the adaptive FEM is given, including assembling stiffness matrix, building load vector, error estimation, and adaptive mesh-refinement. Numerical experiments underline the efficiency of our freely available code which is observed to be of a slightly more than linear complexity with respect to the number of elements when memory limits are not exceeded.
本文论述了在 R d mathbb{R}^{d} （d ∈ N dinmathbb{N} ）中高效实现连续片断线性函数有限元方法（P1-FEM）。虽然目前对使用高维简单分区的有限元方法的实际需求似乎并不高，但研究独立于维数的方法的有效实现还是有一些优势的。例如，它为必要的数据结构和实现的复杂性提供了额外的见解。在整个过程中，重点是使用 Matlab 内置函数和矢量化来高效实现。快速的矢量化 Matlab 函数可以很容易地在许多其他矢量语言中实现，Julia 中也有提供。本文给出了自适应有限元的完整实现方法，包括装配刚度矩阵、构建载荷向量、误差估计和自适应网格细化。数值实验强调了我们免费提供的代码的效率，据观察，在不超过内存限制的情况下，该代码的复杂度略高于元素数量的线性关系。
{"title":"Efficient P1-FEM for Any Space Dimension in Matlab","authors":"Stefanie Beuter, Stefan A. Funken","doi":"10.1515/cmam-2022-0239","DOIUrl":"https://doi.org/10.1515/cmam-2022-0239","url":null,"abstract":"This paper deals with the efficient implementation of the finite element method with continuous piecewise linear functions (P1-FEM) in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"double-struck\">R</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2022-0239_ineq_0001.png\" /> <jats:tex-math>mathbb{R}^{d}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">N</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2022-0239_ineq_0002.png\" /> <jats:tex-math>dinmathbb{N}</jats:tex-math> </jats:alternatives> </jats:inline-formula>). Although at present there does not seem to be a very high practical demand for finite element methods that use higher-dimensional simplicial partitions, there are some advantages in studying the efficient implementation of the method independent of the dimension. For instance, it provides additional insights into necessary data structures and the complexity of implementations. Throughout, the focus is on an efficient realization using <jats:sc>Matlab</jats:sc> built-in functions and vectorization. The fast and vectorized <jats:sc>Matlab</jats:sc> function can be easily implemented in many other vector languages and is provided in Julia, too. The complete implementation of the adaptive FEM is given, including assembling stiffness matrix, building load vector, error estimation, and adaptive mesh-refinement. Numerical experiments underline the efficiency of our freely available code which is observed to be of a slightly more than linear complexity with respect to the number of elements when memory limits are not exceeded.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"7 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557522","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}
In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects. It amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction. To justify the use of Newton’s method for solving this inverse problem, on one hand, we verify well-definedness and differentiability of the forward operator corresponding to two versions of the PDE model; on the other hand, we consider an all-at-once formulation of the inverse problem and prove convergence of Newton’s method for its solution.
{"title":"Simultaneous Reconstruction of Speed of Sound and Nonlinearity Parameter in a Paraxial Model of Vibro-Acoustography in Frequency Domain","authors":"Barbara Kaltenbacher, Teresa Rauscher","doi":"10.1515/cmam-2023-0076","DOIUrl":"https://doi.org/10.1515/cmam-2023-0076","url":null,"abstract":"In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects. It amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction. To justify the use of Newton’s method for solving this inverse problem, on one hand, we verify well-definedness and differentiability of the forward operator corresponding to two versions of the PDE model; on the other hand, we consider an all-at-once formulation of the inverse problem and prove convergence of Newton’s method for its solution.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557530","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}
We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.
{"title":"Robust PRESB Preconditioning of a 3-Dimensional Space-Time Finite Element Method for Parabolic Problems","authors":"Ladislav Foltyn, Dalibor Lukáš, Marco Zank","doi":"10.1515/cmam-2023-0085","DOIUrl":"https://doi.org/10.1515/cmam-2023-0085","url":null,"abstract":"We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"77 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557707","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}
In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term b∈[L∞(Ω)]d{bin[L^{infty}(Omega)]^{d}} and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only H1{H^{1}}-regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. The theoretical results are illustrated by numerical experiments.
本文针对对流项 b∈ [ L ∞ ( Ω ) ] d {bin[L^{infty}(Omega)]^{d}} 的一般二阶非自洽椭圆 PDE，并利用对偶问题的最小正则性（即、对偶问题的解只有 H 1 {H^{1}} -正则性，从而扩展了结果[J.M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer.Anal.46 2008, 5, 2524-2550].数值实验说明了理论结果。
{"title":"Quasi-Optimality of an AFEM for General Second Order Elliptic PDE","authors":"Arnab Pal, Thirupathi Gudi","doi":"10.1515/cmam-2023-0238","DOIUrl":"https://doi.org/10.1515/cmam-2023-0238","url":null,"abstract":"In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>b</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant=\"normal\">∞</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0238_eq_0388.png\" /> <jats:tex-math>{bin[L^{infty}(Omega)]^{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0238_eq_0235.png\" /> <jats:tex-math>{H^{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. The theoretical results are illustrated by numerical experiments.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"36 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139373452","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}
We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances in several numerical experiments.
{"title":"Numerical Approximation of Gaussian Random Fields on Closed Surfaces","authors":"Andrea Bonito, Diane Guignard, Wenyu Lei","doi":"10.1515/cmam-2022-0237","DOIUrl":"https://doi.org/10.1515/cmam-2022-0237","url":null,"abstract":"We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances in several numerical experiments.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"23 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104590","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}
The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can be implemented successfully.
{"title":"An Optimal Method for High-Order Mixed Derivatives of Bivariate Functions","authors":"Evgeniya V. Semenova, Sergiy G. Solodky","doi":"10.1515/cmam-2023-0137","DOIUrl":"https://doi.org/10.1515/cmam-2023-0137","url":null,"abstract":"The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can be implemented successfully.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139079091","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}
This work is concerned with the classical wave equation with a high-contrast coefficient in the spatial derivative operator. We first treat the periodic case, where we derive a new limit in the one-dimensional case. The behavior is illustrated numerically and contrasted to the higher-dimensional case. For general unstructured high-contrast coefficients, we present the Localized Orthogonal Decomposition and show a priori error estimates in suitably weighted norms. Numerical experiments illustrate the convergence rates in various settings.
{"title":"Wave Propagation in High-Contrast Media: Periodic and Beyond","authors":"Élise Fressart, Barbara Verfürth","doi":"10.1515/cmam-2023-0066","DOIUrl":"https://doi.org/10.1515/cmam-2023-0066","url":null,"abstract":"This work is concerned with the classical wave equation with a high-contrast coefficient in the spatial derivative operator. We first treat the periodic case, where we derive a new limit in the one-dimensional case. The behavior is illustrated numerically and contrasted to the higher-dimensional case. For general unstructured high-contrast coefficients, we present the Localized Orthogonal Decomposition and show a priori error estimates in suitably weighted norms. Numerical experiments illustrate the convergence rates in various settings.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"72 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139079450","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}
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