Pub Date : 2024-08-19DOI: 10.1007/s10444-024-10181-0
Bernard Kapidani, Melina Merkel, Sebastian Schöps, Rafael Vázquez
Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.
涡流问题的常见公式涉及矢量或标量电势,两者各有利弊。在存在非三维拓扑的导体时,使用基于标量势的公式会出现僵局。补救办法是用第一同调群的生成器来扩展近似空间。现有的大多数算法都需要特殊的,如分层的有限元基础构造。利用花键的德拉姆复近似的见解,我们证明在这里不需要额外的条件。生成树技术可用于辅助控制网格上的微分形式样条空间和 de Rham 复数之间的同构所产生的六面体网格。
{"title":"Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems","authors":"Bernard Kapidani, Melina Merkel, Sebastian Schöps, Rafael Vázquez","doi":"10.1007/s10444-024-10181-0","DOIUrl":"https://doi.org/10.1007/s10444-024-10181-0","url":null,"abstract":"<p>Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142002702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1007/s10444-024-10185-w
Soura Sana, Bankim C Mandal
This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with different fractional orders. Furthermore, we extend our analysis to encompass the 2D sub-diffusion case. To validate our findings, we conduct numerical experiments to verify the estimated convergence rate. The results confirm the theoretical estimates and provide empirical evidence for the algorithm’s efficiency and reliability. Moreover, we push the boundaries of the algorithm’s applicability by extending it to solve the time fractional Allen-Chan equation, a problem that exceeds our initial theoretical estimates. Remarkably, we observe that the algorithm performs exceptionally well in this extended scenario for both short and long-time windows.
{"title":"Convergence analysis of the Dirichlet-Neumann Waveform Relaxation algorithm for time fractional sub-diffusion and diffusion-wave equations in heterogeneous media","authors":"Soura Sana, Bankim C Mandal","doi":"10.1007/s10444-024-10185-w","DOIUrl":"https://doi.org/10.1007/s10444-024-10185-w","url":null,"abstract":"<p>This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with different fractional orders. Furthermore, we extend our analysis to encompass the 2D sub-diffusion case. To validate our findings, we conduct numerical experiments to verify the estimated convergence rate. The results confirm the theoretical estimates and provide empirical evidence for the algorithm’s efficiency and reliability. Moreover, we push the boundaries of the algorithm’s applicability by extending it to solve the time fractional Allen-Chan equation, a problem that exceeds our initial theoretical estimates. Remarkably, we observe that the algorithm performs exceptionally well in this extended scenario for both short and long-time windows.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141980904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-09DOI: 10.1007/s10444-024-10182-z
Pu-Ting Yu
Let H be a separable Hilbert space and let ({x_{n}}) be a sequence in H that does not contain any zero elements. We say that ({x_{n}}) is a Bessel-normalizable or frame-normalizable sequence if the normalized sequence ({bigl {frac{x_n}{Vert x_nVert }bigr }}) is a Bessel sequence or a frame for H, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs–Stoeva conjecture holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al. as to whether the iterative system (bigl {frac{A^{n} x}{Vert A^{n}xVert }bigr }_{nge 0,, xin S}) associated with a normal operator (A:Hrightarrow H) and a countable subset S of H, is a frame for H. In particular, if S is finite, then we are able to show that (bigl {frac{A^{n} x}{Vert A^{n}xVert }bigr }_{nge 0,, xin S}) is not a frame for H whenever ({A^{n}x}_{nge 0,,xin S}) is a frame for H.
{"title":"Frame-normalizable sequences","authors":"Pu-Ting Yu","doi":"10.1007/s10444-024-10182-z","DOIUrl":"https://doi.org/10.1007/s10444-024-10182-z","url":null,"abstract":"<p>Let <i>H</i> be a separable Hilbert space and let <span>({x_{n}})</span> be a sequence in <i>H</i> that does not contain any zero elements. We say that <span>({x_{n}})</span> is a <i>Bessel-normalizable</i> or <i>frame-normalizable</i> sequence if the normalized sequence <span>({bigl {frac{x_n}{Vert x_nVert }bigr }})</span> is a Bessel sequence or a frame for <i>H</i>, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs–Stoeva conjecture holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al. as to whether the iterative system <span>(bigl {frac{A^{n} x}{Vert A^{n}xVert }bigr }_{nge 0,, xin S})</span> associated with a normal operator <span>(A:Hrightarrow H)</span> and a countable subset <i>S</i> of <i>H</i>, is a frame for <i>H</i>. In particular, if <i>S</i> is finite, then we are able to show that <span>(bigl {frac{A^{n} x}{Vert A^{n}xVert }bigr }_{nge 0,, xin S})</span> is not a frame for <i>H</i> whenever <span>({A^{n}x}_{nge 0,,xin S})</span> is a frame for <i>H</i>.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141908893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-09DOI: 10.1007/s10444-024-10176-x
Anna Yesypenko, Per-Gunnar Martinsson
The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of (varvec{mathcal {H}}^textbf{2})-matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of (varvec{mathcal {H}}) or (varvec{mathcal {H}}^textbf{2}) matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses (varvec{mathcal {H}}^textbf{2})-matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size (textbf{1000} varvec{lambda } times textbf{1000} varvec{lambda }) (for which (varvec{N}~mathbf {=100} textbf{M})) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration.
{"title":"SlabLU: a two-level sparse direct solver for elliptic PDEs","authors":"Anna Yesypenko, Per-Gunnar Martinsson","doi":"10.1007/s10444-024-10176-x","DOIUrl":"https://doi.org/10.1007/s10444-024-10176-x","url":null,"abstract":"<p>The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of <span>(varvec{mathcal {H}}^textbf{2})</span>-matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of <span>(varvec{mathcal {H}})</span> or <span>(varvec{mathcal {H}}^textbf{2})</span> matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses <span>(varvec{mathcal {H}}^textbf{2})</span>-matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size <span>(textbf{1000} varvec{lambda } times textbf{1000} varvec{lambda })</span> (for which <span>(varvec{N}~mathbf {=100} textbf{M})</span>) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141909211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-08DOI: 10.1007/s10444-024-10186-9
Peter Benner, Pawan Goyal
We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21, 143-153 1993). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples.
{"title":"Balanced truncation for quadratic-bilinear control systems","authors":"Peter Benner, Pawan Goyal","doi":"10.1007/s10444-024-10186-9","DOIUrl":"https://doi.org/10.1007/s10444-024-10186-9","url":null,"abstract":"<p>We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. <b>21</b>, 143-153 1993). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141904535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-08DOI: 10.1007/s10444-024-10188-7
Liyao Hao, Wenyi Tian
In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by using the weighted and shifted Grünwald difference (WSGD) formula. Then, a second-order WSGD scheme is obtained after making some initial corrections. Moreover, the error estimates of the proposed time-stepping scheme are rigorously established without the regularity requirement on the exact solution. Finally, some numerical experiments are performed to validate the efficiency and accuracy of the proposed numerical scheme.
{"title":"Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data","authors":"Liyao Hao, Wenyi Tian","doi":"10.1007/s10444-024-10188-7","DOIUrl":"https://doi.org/10.1007/s10444-024-10188-7","url":null,"abstract":"<p>In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by using the weighted and shifted Grünwald difference (WSGD) formula. Then, a second-order WSGD scheme is obtained after making some initial corrections. Moreover, the error estimates of the proposed time-stepping scheme are rigorously established without the regularity requirement on the exact solution. Finally, some numerical experiments are performed to validate the efficiency and accuracy of the proposed numerical scheme.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141904533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-06DOI: 10.1007/s10444-024-10183-y
Ana Alonso Rodríguez, Jessika Camaño, Eduardo De Los Santos, Francesca Rapetti
This paper deals with high order Whitney forms. We define a canonical isomorphism between two sets of degrees of freedom. This allows to geometrically localize the classical degrees of freedom, the moments, over the elements of a simplicial mesh. With such a localization, it is thus possible to associate, even with moments, a graph structure relating a field with its potential.
{"title":"Weights for moments’ geometrical localization: a canonical isomorphism","authors":"Ana Alonso Rodríguez, Jessika Camaño, Eduardo De Los Santos, Francesca Rapetti","doi":"10.1007/s10444-024-10183-y","DOIUrl":"https://doi.org/10.1007/s10444-024-10183-y","url":null,"abstract":"<p>This paper deals with high order Whitney forms. We define a canonical isomorphism between two sets of degrees of freedom. This allows to geometrically localize the classical degrees of freedom, the moments, over the elements of a simplicial mesh. With such a localization, it is thus possible to associate, even with moments, a graph structure relating a field with its potential.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141895604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1007/s10444-024-10167-y
Alessandro Alla, Agnese Pacifico, Michele Palladino, Andrea Pesare
We focus on the control of unknown partial differential equations (PDEs). The system dynamics is unknown, but we assume we are able to observe its evolution for a given control input, as typical in a reinforcement learning framework. We propose an algorithm based on the idea to control and identify on the fly the unknown system configuration. In this work, the control is based on the state-dependent Riccati approach, whereas the identification of the model on Bayesian linear regression. At each iteration, based on the observed data, we obtain an estimate of the a-priori unknown parameter configuration of the PDE and then we compute the control of the correspondent model. We show by numerical evidence the convergence of the method for infinite horizon control problems.
{"title":"Online identification and control of PDEs via reinforcement learning methods","authors":"Alessandro Alla, Agnese Pacifico, Michele Palladino, Andrea Pesare","doi":"10.1007/s10444-024-10167-y","DOIUrl":"https://doi.org/10.1007/s10444-024-10167-y","url":null,"abstract":"<p>We focus on the control of unknown partial differential equations (PDEs). The system dynamics is unknown, but we assume we are able to observe its evolution for a given control input, as typical in a reinforcement learning framework. We propose an algorithm based on the idea to control and identify on the fly the unknown system configuration. In this work, the control is based on the state-dependent Riccati approach, whereas the identification of the model on Bayesian linear regression. At each iteration, based on the observed data, we obtain an estimate of the <i>a-priori</i> unknown parameter configuration of the PDE and then we compute the control of the correspondent model. We show by numerical evidence the convergence of the method for infinite horizon control problems.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-31DOI: 10.1007/s10444-024-10179-8
Mark D. Schubel, Daniel Berwick-Evans, Anil N. Hirani
In exterior calculus on smooth manifolds, the exterior derivative and wedge products are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and an antisymmetrized cup-like product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson’s cochain product defined using Whitney and de Rham maps.
在光滑流形上的外部微积分中,外部导数和楔积对于流形间的光滑映射是自然的,也就是说,这些运算与回拉相通。在离散外部微积分(DEC)中,简单共链扮演了离散形式的角色,共界算子充当了离散外部导数,而反对称杯样积提供了离散楔积。我们证明,DEC 中的这些离散运算对于抽象简单映射是自然的。第二个贡献是对 DEC 中离散楔积的新平均解释。我们还证明了这种楔积与使用惠特尼和德拉姆映射定义的威尔逊共链积是相同的。
{"title":"Averaging property of wedge product and naturality in discrete exterior calculus","authors":"Mark D. Schubel, Daniel Berwick-Evans, Anil N. Hirani","doi":"10.1007/s10444-024-10179-8","DOIUrl":"https://doi.org/10.1007/s10444-024-10179-8","url":null,"abstract":"<p>In exterior calculus on smooth manifolds, the exterior derivative and wedge products are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and an antisymmetrized cup-like product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson’s cochain product defined using Whitney and de Rham maps.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141857629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-30DOI: 10.1007/s10444-024-10180-1
Jing Han, Guangzhi Du
Based on local Gauss integral technique and backtracking technique, this paper presents and studies three kinds of two-grid stabilized finite element algorithms for the stationary Navier-Stokes equations. The proposed methods consist of deducing a coarse solution on the nonlinear system, updating the solution on a fine mesh via three different methods, and solving a linear correction problem on the coarse mesh to obtain the final solution. The error estimates are derived for the solution approximated by the proposed algorithms. A series of numerical experiments are illustrated to test the applicability and efficiency of our proposed methods, and support the theoretical analysis results.
{"title":"Two-grid stabilized finite element methods with backtracking for the stationary Navier-Stokes equations","authors":"Jing Han, Guangzhi Du","doi":"10.1007/s10444-024-10180-1","DOIUrl":"https://doi.org/10.1007/s10444-024-10180-1","url":null,"abstract":"<p>Based on local Gauss integral technique and backtracking technique, this paper presents and studies three kinds of two-grid stabilized finite element algorithms for the stationary Navier-Stokes equations. The proposed methods consist of deducing a coarse solution on the nonlinear system, updating the solution on a fine mesh via three different methods, and solving a linear correction problem on the coarse mesh to obtain the final solution. The error estimates are derived for the solution approximated by the proposed algorithms. A series of numerical experiments are illustrated to test the applicability and efficiency of our proposed methods, and support the theoretical analysis results.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141857628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}