Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint. We design a local discontinuous Galerkin (LDG) method, which imposes a relaxed discrete isometry constraint and controls deformation gradients at barycenters of elements. We prove $varGamma $-convergence of LDG, design a fully practical gradient flow, which gives rise to a linear scheme at every step, and show energy stability and control of the isometry defect. We extend the $varGamma $-convergence analysis to piecewise quadratic creases. We also illustrate the performance of the LDG method with several insightful simulations of large deformations, one including a curved crease.
{"title":"Gamma-convergent LDG method for large bending deformations of bilayer plates","authors":"Andrea Bonito, Ricardo H Nochetto, Shuo Yang","doi":"10.1093/imanum/drad100","DOIUrl":"https://doi.org/10.1093/imanum/drad100","url":null,"abstract":"Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint. We design a local discontinuous Galerkin (LDG) method, which imposes a relaxed discrete isometry constraint and controls deformation gradients at barycenters of elements. We prove $varGamma $-convergence of LDG, design a fully practical gradient flow, which gives rise to a linear scheme at every step, and show energy stability and control of the isometry defect. We extend the $varGamma $-convergence analysis to piecewise quadratic creases. We also illustrate the performance of the LDG method with several insightful simulations of large deformations, one including a curved crease.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139505921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alejandro Allendes, Gilberto Campaña, Francisco Fuica, Enrique Otárola
We study the existence of solutions for Darcy’s problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The model studied involves thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case where thermal diffusion is independent of temperature, we propose an a posteriori error estimator and study its reliability and efficiency properties. We illustrate the theory with numerical examples.
{"title":"Darcy’s problem coupled with the heat equation under singular forcing: analysis and discretization","authors":"Alejandro Allendes, Gilberto Campaña, Francisco Fuica, Enrique Otárola","doi":"10.1093/imanum/drad094","DOIUrl":"https://doi.org/10.1093/imanum/drad094","url":null,"abstract":"We study the existence of solutions for Darcy’s problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The model studied involves thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case where thermal diffusion is independent of temperature, we propose an a posteriori error estimator and study its reliability and efficiency properties. We illustrate the theory with numerical examples.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139431258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function $f(k(x,t),u)$, where the coefficient $k(x,t)$ is $BV$-regular and may exhibit discontinuities along curves in the $(x,t)$ plane. Our approach, which is technically elementary, relies on a discrete interaction estimate and one entropy function. While the details are specifically outlined for the Lax-Friedrichs scheme, the same framework can be applied to other difference schemes. Notably, our compactness estimates are new even in the homogeneous case ($kequiv 1$).
{"title":"Compactness estimates for difference schemes for conservation laws with discontinuous flux","authors":"Kenneth H Karlsen, John D Towers","doi":"10.1093/imanum/drad096","DOIUrl":"https://doi.org/10.1093/imanum/drad096","url":null,"abstract":"We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function $f(k(x,t),u)$, where the coefficient $k(x,t)$ is $BV$-regular and may exhibit discontinuities along curves in the $(x,t)$ plane. Our approach, which is technically elementary, relies on a discrete interaction estimate and one entropy function. While the details are specifically outlined for the Lax-Friedrichs scheme, the same framework can be applied to other difference schemes. Notably, our compactness estimates are new even in the homogeneous case ($kequiv 1$).","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139101268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems over the Stiefel manifold. We propose a framework for developing subgradient-based methods and establishing their convergence properties based on prior works. Furthermore, based on our proposed framework, we can develop efficient approaches for optimization over the Stiefel manifold. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach yields efficient and direct implementations of various unconstrained approaches to nonsmooth optimization problems over the Stiefel manifold.
{"title":"A constraint dissolving approach for nonsmooth optimization over the Stiefel manifold","authors":"Xiaoyin Hu, Nachuan Xiao, Xin Liu, Kim-Chuan Toh","doi":"10.1093/imanum/drad098","DOIUrl":"https://doi.org/10.1093/imanum/drad098","url":null,"abstract":"This paper focuses on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems over the Stiefel manifold. We propose a framework for developing subgradient-based methods and establishing their convergence properties based on prior works. Furthermore, based on our proposed framework, we can develop efficient approaches for optimization over the Stiefel manifold. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach yields efficient and direct implementations of various unconstrained approaches to nonsmooth optimization problems over the Stiefel manifold.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139041468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we derive a solution to the problem of finding a polynomial of degree at most $n$ that best approximates data at $n+2$ points in the $l_{p}$ norm. Analogous to a result of de la Vallée Poussin, one can express the solution as a convex combination of the Lagrange interpolants over subsets of $n+1$ points, and the error oscillates in sign.
在本论文中，我们推导出了一个问题的解决方案，即找到一个度数最多为 $n$ 的多项式，该多项式在 $l_{p}$ 准则下最接近 $n+2$ 点的数据。与 de la Vallée Poussin 的一个结果类似，我们可以将解表示为 $n+1$ 点子集上的拉格朗日内插值的凸组合，并且误差在符号上摆动。
{"title":"On best p-norm approximation of discrete data by polynomials","authors":"Michael S Floater","doi":"10.1093/imanum/drad086","DOIUrl":"https://doi.org/10.1093/imanum/drad086","url":null,"abstract":"In this note, we derive a solution to the problem of finding a polynomial of degree at most $n$ that best approximates data at $n+2$ points in the $l_{p}$ norm. Analogous to a result of de la Vallée Poussin, one can express the solution as a convex combination of the Lagrange interpolants over subsets of $n+1$ points, and the error oscillates in sign.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138559326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze the convergence properties of discretizations and obtain, in two dimensions, an a priori error estimate for the underlying approximation of an optimal control variable.
{"title":"Semilinear optimal control with Dirac measures","authors":"Enrique Otárola","doi":"10.1093/imanum/drad091","DOIUrl":"https://doi.org/10.1093/imanum/drad091","url":null,"abstract":"The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze the convergence properties of discretizations and obtain, in two dimensions, an a priori error estimate for the underlying approximation of an optimal control variable.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138455384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive new multiple point flux approximations (MPFA) for the finite volume approximation of heterogeneous and anisotropic diffusion problems on general meshes, in dimensions 2 and 3. The resulting methods are unconditionally stable while preserving the small stencil typical of MPFA finite volumes. The key idea is to solve local variational problems with a well-designed stabilization term from which we deduce conservative flux instead of directly prescribing a flux formula and solving the usual flux continuity equations. The boundary conditions of our local variational problems are handled through additional cell-centered unknowns, leading to an overall scheme with the same number of unknowns than first-order discontinuous Galerkin methods. Convergence results follow from well-established frameworks, while numerical experiments illustrate the good behavior of the method.
{"title":"Unconditionally stable small stencil enriched multiple point flux approximations of heterogeneous diffusion problems on general meshes","authors":"Julien Coatléven","doi":"10.1093/imanum/drad087","DOIUrl":"https://doi.org/10.1093/imanum/drad087","url":null,"abstract":"We derive new multiple point flux approximations (MPFA) for the finite volume approximation of heterogeneous and anisotropic diffusion problems on general meshes, in dimensions 2 and 3. The resulting methods are unconditionally stable while preserving the small stencil typical of MPFA finite volumes. The key idea is to solve local variational problems with a well-designed stabilization term from which we deduce conservative flux instead of directly prescribing a flux formula and solving the usual flux continuity equations. The boundary conditions of our local variational problems are handled through additional cell-centered unknowns, leading to an overall scheme with the same number of unknowns than first-order discontinuous Galerkin methods. Convergence results follow from well-established frameworks, while numerical experiments illustrate the good behavior of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138449679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
L Beirão da Veiga, C Canuto, R H Nochetto, G Vacca, M Verani
We design an adaptive virtual element method (AVEM) of lowest order over triangular meshes with hanging nodes in 2d, which are treated as polygons. AVEM hinges on the stabilization-free a posteriori error estimators recently derived in Beirão da Veiga et al. (2023, Adaptive VEM: stabilization-free a posteriori error analysis and contraction property. SIAM J. Numer. Anal., 61, 457–494). The crucial property, which also plays a central role in this paper, is that the stabilization term can be made arbitrarily small relative to the a posteriori error estimators upon increasing the stabilization parameter. Our AVEM concatenates two modules, GALERKIN and DATA. The former deals with piecewise constant data and is shown in the above article to be a contraction between consecutive iterates. The latter approximates general data by piecewise constants to a desired accuracy. AVEM is shown to be convergent and quasi-optimal, in terms of error decay versus degrees of freedom, for solutions and data belonging to appropriate approximation classes. Numerical experiments illustrate the interplay between these two modules and provide computational evidence of optimality.
本文设计了一种最低阶自适应虚元法(AVEM)，将二维三角形网格作为多边形处理。AVEM依赖于最近在beir o da Veiga等人(2023)中导出的无稳定化后检误差估计，自适应VEM:无稳定化后检误差分析和收缩特性。SIAM J. number。分析的， 61, 457-494)。关键的性质，也是在本文中起中心作用，是稳定项可以使任意小相对于后检误差估计在增加稳定参数。我们的AVEM连接两个模块，GALERKIN和DATA。前者处理分段常量数据，并在上面的文章中显示为连续迭代之间的收缩。后者通过分段常数逼近一般数据，达到所需的精度。对于属于适当近似类的解和数据，AVEM被证明是收敛的和准最优的，就误差衰减与自由度而言。数值实验说明了这两个模块之间的相互作用，并提供了最优性的计算证据。
{"title":"Adaptive VEM for variable data: convergence and optimality","authors":"L Beirão da Veiga, C Canuto, R H Nochetto, G Vacca, M Verani","doi":"10.1093/imanum/drad085","DOIUrl":"https://doi.org/10.1093/imanum/drad085","url":null,"abstract":"We design an adaptive virtual element method (AVEM) of lowest order over triangular meshes with hanging nodes in 2d, which are treated as polygons. AVEM hinges on the stabilization-free a posteriori error estimators recently derived in Beirão da Veiga et al. (2023, Adaptive VEM: stabilization-free a posteriori error analysis and contraction property. SIAM J. Numer. Anal., 61, 457–494). The crucial property, which also plays a central role in this paper, is that the stabilization term can be made arbitrarily small relative to the a posteriori error estimators upon increasing the stabilization parameter. Our AVEM concatenates two modules, GALERKIN and DATA. The former deals with piecewise constant data and is shown in the above article to be a contraction between consecutive iterates. The latter approximates general data by piecewise constants to a desired accuracy. AVEM is shown to be convergent and quasi-optimal, in terms of error decay versus degrees of freedom, for solutions and data belonging to appropriate approximation classes. Numerical experiments illustrate the interplay between these two modules and provide computational evidence of optimality.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zorica Milovanović Jeknić, Aleksandra Delić, Sandra Živanović
We consider an elliptic boundary value problem with nonlocal conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, is obtained.
{"title":"A two-dimensional boundary value problem of elliptic type with nonlocal conjugation conditions","authors":"Zorica Milovanović Jeknić, Aleksandra Delić, Sandra Živanović","doi":"10.1093/imanum/drad084","DOIUrl":"https://doi.org/10.1093/imanum/drad084","url":null,"abstract":"We consider an elliptic boundary value problem with nonlocal conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, is obtained.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"109126944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity problems with close-to-touching inclusions of general geometry still remains open in three dimensions. We address this problem by establishing new asymptotic estimates for the second-order partial derivatives of the solution with explicit dependence on the distance $varepsilon $ between the inclusions, and use the asymptotic estimates to design a class of graded meshes and finite element spaces to solve the perfect conductivity problem with possibly close-to-touching inclusions. In particular, we propose a special finite element basis function that resolves the asymptotic singularity of the solution by making the interpolation error bounded in $W^{1,infty }$ in a neighborhood of the close-to-touching point, even though the solution itself is blowing up in $W^{1,infty }$. This is crucial in the error analysis for the numerical approximations. We prove that the proposed method yields optimal-order convergence in the $H^1$ norm, uniformly with respect to the distance $varepsilon $ between the inclusions, in both two and three dimensions for general convex smooth inclusions, which are possibly close-to-touching. Numerical experiments are presented to support the theoretical analysis and to illustrate the convergence of the proposed method for different shapes of inclusions in both two- and three-dimensional domains.
{"title":"Convergent finite element methods for the perfect conductivity problem with close-to-touching inclusions","authors":"Buyang Li, Haigang Li, Zongze Yang","doi":"10.1093/imanum/drad088","DOIUrl":"https://doi.org/10.1093/imanum/drad088","url":null,"abstract":"In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity problems with close-to-touching inclusions of general geometry still remains open in three dimensions. We address this problem by establishing new asymptotic estimates for the second-order partial derivatives of the solution with explicit dependence on the distance $varepsilon $ between the inclusions, and use the asymptotic estimates to design a class of graded meshes and finite element spaces to solve the perfect conductivity problem with possibly close-to-touching inclusions. In particular, we propose a special finite element basis function that resolves the asymptotic singularity of the solution by making the interpolation error bounded in $W^{1,infty }$ in a neighborhood of the close-to-touching point, even though the solution itself is blowing up in $W^{1,infty }$. This is crucial in the error analysis for the numerical approximations. We prove that the proposed method yields optimal-order convergence in the $H^1$ norm, uniformly with respect to the distance $varepsilon $ between the inclusions, in both two and three dimensions for general convex smooth inclusions, which are possibly close-to-touching. Numerical experiments are presented to support the theoretical analysis and to illustrate the convergence of the proposed method for different shapes of inclusions in both two- and three-dimensional domains.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"109126945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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