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.
{"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.
{"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.
{"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}
Optimal-order convergence in the $H^{1}$ norm is proved for an arbitrary Lagrangian–Eulerian (ALE) interface tracking finite element method (FEM) for the sharp interface model of two-phase Navier–Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid’s velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete ALE interface tracking FEM is shown to be $O(h^{k})$ in the $L^infty (0, T; H^{1}(varOmega ))$ norm for the Taylor–Hood finite elements of degree $k geqslant 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.
{"title":"Optimal convergence of the arbitrary Lagrangian–Eulerian interface tracking method for two-phase Navier–Stokes flow without surface tension","authors":"Buyang Li, Shu Ma, Weifeng Qiu","doi":"10.1093/imanum/draf003","DOIUrl":"https://doi.org/10.1093/imanum/draf003","url":null,"abstract":"Optimal-order convergence in the $H^{1}$ norm is proved for an arbitrary Lagrangian–Eulerian (ALE) interface tracking finite element method (FEM) for the sharp interface model of two-phase Navier–Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid’s velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete ALE interface tracking FEM is shown to be $O(h^{k})$ in the $L^infty (0, T; H^{1}(varOmega ))$ norm for the Taylor–Hood finite elements of degree $k geqslant 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143736499","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 construct and analyse finite element approximations of the Einstein tensor in dimension $N ge 3$. We focus on the setting where a smooth Riemannian metric tensor $g$ on a polyhedral domain $varOmega subset mathbb{R}^{N}$ has been approximated by a piecewise polynomial metric $g_{h}$ on a simplicial triangulation $mathcal{T}$ of $varOmega $ having maximum element diameter $h$. We assume that $g_{h}$ possesses single-valued tangential–tangential components on every codimension-$1$ simplex in $mathcal{T}$. Such a metric is not classically differentiable in general, but it turns out that one can still attribute meaning to its Einstein curvature in a distributional sense. We study the convergence of the distributional Einstein curvature of $g_{h}$ to the Einstein curvature of $g$ under refinement of the triangulation. We show that in the $H^{-2}(varOmega )$-norm this convergence takes place at a rate of $O(h^{r+1})$ when $g_{h}$ is an optimal-order interpolant of $g$ that is piecewise polynomial of degree $r ge 1$. We provide numerical evidence to support this claim. In the process of proving our convergence results we derive a few formulas for the evolution of certain geometric quantities under deformations of the metric.
{"title":"Finite element approximation of the Einstein tensor","authors":"Evan S Gawlik, Michael Neunteufel","doi":"10.1093/imanum/draf004","DOIUrl":"https://doi.org/10.1093/imanum/draf004","url":null,"abstract":"We construct and analyse finite element approximations of the Einstein tensor in dimension $N ge 3$. We focus on the setting where a smooth Riemannian metric tensor $g$ on a polyhedral domain $varOmega subset mathbb{R}^{N}$ has been approximated by a piecewise polynomial metric $g_{h}$ on a simplicial triangulation $mathcal{T}$ of $varOmega $ having maximum element diameter $h$. We assume that $g_{h}$ possesses single-valued tangential–tangential components on every codimension-$1$ simplex in $mathcal{T}$. Such a metric is not classically differentiable in general, but it turns out that one can still attribute meaning to its Einstein curvature in a distributional sense. We study the convergence of the distributional Einstein curvature of $g_{h}$ to the Einstein curvature of $g$ under refinement of the triangulation. We show that in the $H^{-2}(varOmega )$-norm this convergence takes place at a rate of $O(h^{r+1})$ when $g_{h}$ is an optimal-order interpolant of $g$ that is piecewise polynomial of degree $r ge 1$. We provide numerical evidence to support this claim. In the process of proving our convergence results we derive a few formulas for the evolution of certain geometric quantities under deformations of the metric.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"72 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143736497","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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