In this paper, we describe an algorithm for approximating functions of the form $f(x)=int _{a}^{b} x^{mu } sigma (mu ) , {text{d}} mu $ over $[0,1]$, where $sigma (mu )$ is some signed Radon measure, or, more generally, of the form $f(x) = {{langle sigma (mu ), x^mu rangle }}$, where $sigma (mu )$ is some distribution supported on $[a,b]$, with $0 <a < b< infty $. One example from this class of functions is $x^{c} (log{x})^{m}=(-1)^{m} {{langle delta ^{(m)}(mu -c), x^mu rangle }}$, where $aleq c leq b$ and $m geq 0$ is an integer. Given the desired accuracy $varepsilon $ and the values of $a$ and $b$, our method determines a priori a collection of noninteger powers $t_{1}$, $t_{2}$, …, $t_{N}$, so that the functions are approximated by series of the form $f(x)approx sum _{j=1}^{N} c_{j} x^{t_{j}}$, and a set of collocation points $x_{1}$, $x_{2}$, …, $x_{N}$, such that the expansion coefficients can be found by collocating the function at these points. We prove that our method has a small uniform approximation error, which is proportional to $varepsilon $ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(log{frac{1}{varepsilon }})$. We demonstrate the performance of our algorithm with several numerical experiments.
{"title":"On the approximation of singular functions by series of noninteger powers","authors":"Mohan Zhao, Kirill Serkh","doi":"10.1093/imanum/draf006","DOIUrl":"https://doi.org/10.1093/imanum/draf006","url":null,"abstract":"In this paper, we describe an algorithm for approximating functions of the form $f(x)=int _{a}^{b} x^{mu } sigma (mu ) , {text{d}} mu $ over $[0,1]$, where $sigma (mu )$ is some signed Radon measure, or, more generally, of the form $f(x) = {{langle sigma (mu ), x^mu rangle }}$, where $sigma (mu )$ is some distribution supported on $[a,b]$, with $0 &lt;a &lt; b&lt; infty $. One example from this class of functions is $x^{c} (log{x})^{m}=(-1)^{m} {{langle delta ^{(m)}(mu -c), x^mu rangle }}$, where $aleq c leq b$ and $m geq 0$ is an integer. Given the desired accuracy $varepsilon $ and the values of $a$ and $b$, our method determines a priori a collection of noninteger powers $t_{1}$, $t_{2}$, …, $t_{N}$, so that the functions are approximated by series of the form $f(x)approx sum _{j=1}^{N} c_{j} x^{t_{j}}$, and a set of collocation points $x_{1}$, $x_{2}$, …, $x_{N}$, such that the expansion coefficients can be found by collocating the function at these points. We prove that our method has a small uniform approximation error, which is proportional to $varepsilon $ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(log{frac{1}{varepsilon }})$. We demonstrate the performance of our algorithm with several numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143878121","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 consider a hybrid FEM-BEM method to compute approximations of full-space linear elliptic transmission problems. First, we derive a priori and a posteriori error estimates. Then, building on the latter, we present an adaptive algorithm and prove that it converges at optimal rates with respect to the number of mesh elements. Finally, we provide numerical experiments, demonstrating the practical performance of the adaptive algorithm.
{"title":"Optimal convergence rates of an adaptive hybrid FEM-BEM method for full-space linear transmission problems","authors":"Gregor Gantner, Michele Ruggeri","doi":"10.1093/imanum/draf023","DOIUrl":"https://doi.org/10.1093/imanum/draf023","url":null,"abstract":"We consider a hybrid FEM-BEM method to compute approximations of full-space linear elliptic transmission problems. First, we derive a priori and a posteriori error estimates. Then, building on the latter, we present an adaptive algorithm and prove that it converges at optimal rates with respect to the number of mesh elements. Finally, we provide numerical experiments, demonstrating the practical performance of the adaptive algorithm.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143875731","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 analyze a space-time hybridizable discontinuous Galerkin method to solve the time-dependent advection-diffusion equation. We prove stability of the discretization in the advection-dominated regime by using weighted test functions and derive a priori space-time error estimates. Numerical examples illustrate the theoretical results.
{"title":"Space-time hybridizable discontinuous Galerkin method for advection-diffusion: the advection-dominated regime","authors":"Yuan Wang, Sander Rhebergen","doi":"10.1093/imanum/draf013","DOIUrl":"https://doi.org/10.1093/imanum/draf013","url":null,"abstract":"We analyze a space-time hybridizable discontinuous Galerkin method to solve the time-dependent advection-diffusion equation. We prove stability of the discretization in the advection-dominated regime by using weighted test functions and derive a priori space-time error estimates. Numerical examples illustrate the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143866498","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 present and analyze a hybridizable discontinuous Galerkin method for coupling Stokes and Darcy equations, whose domains are discretized by two independent triangulations. This causes nonconformity at the intersection of the subdomains or leaves a gap (unmeshed region) between them. In order to properly couple the two different discretizations and obtain a high-order scheme, we propose suitable transmission conditions based on mass conservation, equilibrium of normal forces and the Beavers–Joseph–Saffman law. Since the meshes do not necessarily coincide, we use the Transfer Path Method to tie them. We establish the well-posedness of the method and provide error estimates where the influences of the nonconformity and the gap are explicit in the constants. Finally, numerical experiments that illustrate the performance of the method are shown.
提出并分析了Stokes方程和Darcy方程耦合的一种可杂交不连续Galerkin方法，该方法的域由两个独立的三角剖分离散。这导致子域相交处的不整合或在它们之间留下间隙（未网格区域）。在质量守恒、法向力平衡和beaver - joseph - saffman定律的基础上，提出了合适的传输条件，使两种不同的离散化得到高阶格式。由于网格不一定重合，我们使用传输路径方法来连接它们。我们建立了该方法的适定性，并提供了误差估计，其中不一致性和间隙的影响在常数中是显式的。最后，通过数值实验验证了该方法的有效性。
{"title":"A hybridizable discontinuous Galerkin method for Stokes/Darcy coupling on dissimilar meshes","authors":"Isaac Bermúdez, Jaime Manríquez, Manuel Solano","doi":"10.1093/imanum/drae109","DOIUrl":"https://doi.org/10.1093/imanum/drae109","url":null,"abstract":"We present and analyze a hybridizable discontinuous Galerkin method for coupling Stokes and Darcy equations, whose domains are discretized by two independent triangulations. This causes nonconformity at the intersection of the subdomains or leaves a gap (unmeshed region) between them. In order to properly couple the two different discretizations and obtain a high-order scheme, we propose suitable transmission conditions based on mass conservation, equilibrium of normal forces and the Beavers–Joseph–Saffman law. Since the meshes do not necessarily coincide, we use the Transfer Path Method to tie them. We establish the well-posedness of the method and provide error estimates where the influences of the nonconformity and the gap are explicit in the constants. Finally, numerical experiments that illustrate the performance of the method are shown.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143851018","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 paper, we consider a state constrained optimal control problem governed by the transient Stokes equations. The state constraint is given by an $L^{2}$ functional in space, which is required to fulfill a pointwise bound in time. The discretization scheme for the Stokes equations consists of inf-sup stable finite elements in space and a discontinuous Galerkin method in time, for which we have recently established best approximation type error estimates. Using these error estimates, for the discrete control problem we derive error estimates and as a by-product we show an improved regularity for the optimal control. We complement our theoretical analysis with numerical results.
{"title":"A priori error estimates for optimal control problems governed by the transient Stokes equations and subject to state constraints pointwise in time","authors":"Dmitriy Leykekhman, Boris Vexler, Jakob Wagner","doi":"10.1093/imanum/draf018","DOIUrl":"https://doi.org/10.1093/imanum/draf018","url":null,"abstract":"In this paper, we consider a state constrained optimal control problem governed by the transient Stokes equations. The state constraint is given by an $L^{2}$ functional in space, which is required to fulfill a pointwise bound in time. The discretization scheme for the Stokes equations consists of inf-sup stable finite elements in space and a discontinuous Galerkin method in time, for which we have recently established best approximation type error estimates. Using these error estimates, for the discrete control problem we derive error estimates and as a by-product we show an improved regularity for the optimal control. We complement our theoretical analysis with numerical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143824844","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 consider a general family of nonlocal in space and time diffusion equations with space-time dependent diffusivity and prove convergence of finite difference schemes in the context of viscosity solutions under very mild conditions. The proofs, based on regularity properties and compactness arguments on the numerical solution, allow to inherit a number of interesting results for the limit equation. More precisely, assuming Hölder regularity only on the initial condition, we prove convergence of the scheme, space-time Hölder regularity of the solution, depending on the fractional orders of the operators, as well as specific blow up rates of the first time derivative. The schemes’ performance is further numerically verified using both constructed exact solutions and realistic examples. Our experiments show that multithreaded implementation yields an efficient method to solve nonlocal equations numerically.
{"title":"Numerical methods and regularity properties for viscosity solutions of nonlocal in space and time diffusion equations","authors":"Félix del Teso, Łukasz Płociniczak","doi":"10.1093/imanum/draf011","DOIUrl":"https://doi.org/10.1093/imanum/draf011","url":null,"abstract":"We consider a general family of nonlocal in space and time diffusion equations with space-time dependent diffusivity and prove convergence of finite difference schemes in the context of viscosity solutions under very mild conditions. The proofs, based on regularity properties and compactness arguments on the numerical solution, allow to inherit a number of interesting results for the limit equation. More precisely, assuming Hölder regularity only on the initial condition, we prove convergence of the scheme, space-time Hölder regularity of the solution, depending on the fractional orders of the operators, as well as specific blow up rates of the first time derivative. The schemes’ performance is further numerically verified using both constructed exact solutions and realistic examples. Our experiments show that multithreaded implementation yields an efficient method to solve nonlocal equations numerically.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143813720","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 paper a staggered mixed method based on the first-order system is proposed and analysed. The proposed method uses piecewise polynomial spaces enjoying partial continuity properties and hinges on a careful balancing of the involved finite element spaces. It supports polygonal meshes of arbitrary shapes and is free of stabilization. All the variables converge optimally with respect to the polynomial order as demonstrated by the rigorous convergence analysis. In particular, it is shown that the approximations of $u$ and $boldsymbol{p}:=nabla u$ superconverge to the suitably defined projections, and it is noteworthy that the approximation of $u$ superconverges to the projection in $L^{2}$-error of order $k+3$ up to the data oscillation term when polynomials of degree $k-1$ are used for the approximation of $u$. Taking advantage of the superconvergence we are able to define the local postprocessing approximations for $u$ and $boldsymbol{p}$, respectively. The convergence error estimates for the postprocessing approximations are also proved. Several numerical experiments are presented to confirm the proposed theories.
{"title":"A staggered mixed method for the biharmonic problem based on the first-order system","authors":"Lina Zhao","doi":"10.1093/imanum/draf021","DOIUrl":"https://doi.org/10.1093/imanum/draf021","url":null,"abstract":"In this paper a staggered mixed method based on the first-order system is proposed and analysed. The proposed method uses piecewise polynomial spaces enjoying partial continuity properties and hinges on a careful balancing of the involved finite element spaces. It supports polygonal meshes of arbitrary shapes and is free of stabilization. All the variables converge optimally with respect to the polynomial order as demonstrated by the rigorous convergence analysis. In particular, it is shown that the approximations of $u$ and $boldsymbol{p}:=nabla u$ superconverge to the suitably defined projections, and it is noteworthy that the approximation of $u$ superconverges to the projection in $L^{2}$-error of order $k+3$ up to the data oscillation term when polynomials of degree $k-1$ are used for the approximation of $u$. Taking advantage of the superconvergence we are able to define the local postprocessing approximations for $u$ and $boldsymbol{p}$, respectively. The convergence error estimates for the postprocessing approximations are also proved. Several numerical experiments are presented to confirm the proposed theories.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"37 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797795","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}
A proof of optimal-order error estimates is given for the full discretization of the Cahn–Hilliard equation with Cahn–Hilliard-type dynamic boundary conditions in a smooth domain. The numerical method combines a linear bulk–surface finite element discretization in space and linearly implicit backward difference formulae of order 1 to 5 in time. Optimal-order error estimates are proven. The error estimates are based on a consistency and stability analysis in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second-order system.
{"title":"Error estimates for full discretization of Cahn–Hilliard equation with dynamic boundary conditions","authors":"Nils Bullerjahn, Balázs Kovács","doi":"10.1093/imanum/draf009","DOIUrl":"https://doi.org/10.1093/imanum/draf009","url":null,"abstract":"A proof of optimal-order error estimates is given for the full discretization of the Cahn–Hilliard equation with Cahn–Hilliard-type dynamic boundary conditions in a smooth domain. The numerical method combines a linear bulk–surface finite element discretization in space and linearly implicit backward difference formulae of order 1 to 5 in time. Optimal-order error estimates are proven. The error estimates are based on a consistency and stability analysis in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second-order system.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143782663","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 randomized unbiased estimators of Rhee & Glynn (2015, Unbiased estimation with square root convergence for SDE models. Oper. Res, 63, 1026–1043) can be highly efficient at approximating expectations of path functionals associated with stochastic differential equations. However, algorithms for calculating the optimal distributions with an infinite horizon are lacking. In this article, based on the method of Cui et al. (2021, On the optimal design of the randomized unbiased Monte Carlo estimators. Oper. Res. Lett., 49, 477–484), we prove that, under mild assumptions, there is a simple representation of the optimal distributions. Then, we develop an adaptive algorithm to compute the optimal distributions with an infinite horizon, which requires only a small amount of computational time in prior estimation. Finally, we provide numerical results to illustrate the efficiency of our adaptive algorithm.
随机无偏估计量Rhee &；Glynn (2015)， SDE模型的平方根收敛无偏估计。③。Res, 63, 1026-1043)可以高效地逼近与随机微分方程相关的路径函数的期望。然而，计算无限视界下的最优分布的算法是缺乏的。本文基于Cui et al.（2021）的方法，研究随机无偏蒙特卡罗估计器的优化设计。③。卷。在温和的假设下，我们证明了存在最优分布的简单表示。然后，我们开发了一种自适应算法来计算具有无限视界的最优分布，该算法只需要少量的先验估计计算时间。最后，给出了数值结果来说明自适应算法的有效性。
{"title":"Optimal distributions for randomized unbiased estimators with an infinite horizon and an adaptive algorithm","authors":"Chao Zheng, Jiangtao Pan, Qun Wang","doi":"10.1093/imanum/draf017","DOIUrl":"https://doi.org/10.1093/imanum/draf017","url":null,"abstract":"The randomized unbiased estimators of Rhee & Glynn (2015, Unbiased estimation with square root convergence for SDE models. Oper. Res, 63, 1026–1043) can be highly efficient at approximating expectations of path functionals associated with stochastic differential equations. However, algorithms for calculating the optimal distributions with an infinite horizon are lacking. In this article, based on the method of Cui et al. (2021, On the optimal design of the randomized unbiased Monte Carlo estimators. Oper. Res. Lett., 49, 477–484), we prove that, under mild assumptions, there is a simple representation of the optimal distributions. Then, we develop an adaptive algorithm to compute the optimal distributions with an infinite horizon, which requires only a small amount of computational time in prior estimation. Finally, we provide numerical results to illustrate the efficiency of our adaptive algorithm.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"73 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143775386","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 work, we are interested in a class of numerical schemes for certain phase field models. It is well known that unconditional energy stability (energy decays in time regardless of the size of the time step) provides a fidelity check in practical numerical simulations. In recent work (Li, D. (2022b, Why large time-stepping methods for the Cahn–Hilliard equation is stable. Math. Comp., 91, 2501–2515)), a type of semi-implicit scheme for the Cahn–Hilliard (CH) equation with regular potential was developed satisfying the energy-decay property. In this paper, we extend such semi-implicit schemes to the Allen–Cahn equation and the fractional CH equation with a rigorous proof of similar energy stability. Models in two spatial dimensions are discussed.
在这项工作中，我们感兴趣的是一类特定相场模型的数值格式。众所周知，无条件能量稳定性（能量随时间衰减而与时间步长无关）在实际数值模拟中提供了保真度检查。在最近的工作中(Li， D. (2022b)，为什么大时间步进方法求解Cahn-Hilliard方程是稳定的。数学。（p., 91, 2501-2515）)，给出了一种满足能量衰减性质的具有规则势的Cahn-Hilliard （CH）方程的半隐式格式。本文将这种半隐式格式推广到Allen-Cahn方程和分数阶CH方程，并给出了类似能量稳定性的严格证明。讨论了两个空间维度上的模型。
{"title":"Energy stable semi-implicit schemes for the 2D Allen–Cahn and fractional Cahn–Hilliard equations","authors":"Xinyu Cheng","doi":"10.1093/imanum/draf010","DOIUrl":"https://doi.org/10.1093/imanum/draf010","url":null,"abstract":"In this work, we are interested in a class of numerical schemes for certain phase field models. It is well known that unconditional energy stability (energy decays in time regardless of the size of the time step) provides a fidelity check in practical numerical simulations. In recent work (Li, D. (2022b, Why large time-stepping methods for the Cahn–Hilliard equation is stable. Math. Comp., 91, 2501–2515)), a type of semi-implicit scheme for the Cahn–Hilliard (CH) equation with regular potential was developed satisfying the energy-decay property. In this paper, we extend such semi-implicit schemes to the Allen–Cahn equation and the fractional CH equation with a rigorous proof of similar energy stability. Models in two spatial dimensions are discussed.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"72 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143744936","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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