We present an error analysis of weak convergence of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. Following Milstein’s weak error analysis on the one-step approximation of SDEs, we prove a general result on weak convergence of the one-step discretization of the SDEs mentioned above. As applications, we show the weak convergence rates for several numerical schemes of half-order strong convergence, such as tamed and balanced schemes. Numerical examples are presented to verify our theoretical analysis.
{"title":"Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients","authors":"Xiaojie Wang, Yuying Zhao, Zhongqiang Zhang","doi":"10.1093/imanum/drad083","DOIUrl":"https://doi.org/10.1093/imanum/drad083","url":null,"abstract":"We present an error analysis of weak convergence of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. Following Milstein’s weak error analysis on the one-step approximation of SDEs, we prove a general result on weak convergence of the one-step discretization of the SDEs mentioned above. As applications, we show the weak convergence rates for several numerical schemes of half-order strong convergence, such as tamed and balanced schemes. Numerical examples are presented to verify our theoretical analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"109126943","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 mixed form of the Cahn–Hilliard equations is discretized by the hybridized discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg–Landau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo–Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces.
{"title":"Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn–Hilliard problem","authors":"Keegan L A Kirk, Beatrice Riviere, Rami Masri","doi":"10.1093/imanum/drad075","DOIUrl":"https://doi.org/10.1093/imanum/drad075","url":null,"abstract":"The mixed form of the Cahn–Hilliard equations is discretized by the hybridized discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg–Landau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo–Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"109127013","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 propose a monotone and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized infinity Laplacian, which could be related to the family of the so-called two-scale methods. We show that this method is convergent and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right-hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms, are also considered.
{"title":"Two-scale methods for the normalized infinity Laplacian: rates of convergence","authors":"Wenbo Li, Abner J Salgado","doi":"10.1093/imanum/drad074","DOIUrl":"https://doi.org/10.1093/imanum/drad074","url":null,"abstract":"We propose a monotone and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized infinity Laplacian, which could be related to the family of the so-called two-scale methods. We show that this method is convergent and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right-hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms, are also considered.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"109126952","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 an energy-conserving continuous Galerkin discretization of the Gross–Pitaevskii equation with a magnetic trapping potential and a stirring potential for angular momentum rotation. The discretization is based on finite elements in space and time and allows for arbitrary polynomial orders. It was first analyzed by O. Karakashian and C. Makridakis (SIAM J. Numer. Anal., 36(6),1779–1807, 1999) in the absence of potential terms and corresponding a priori error estimates were derived in $2D$. In this work we revisit the approach in the generalized setting of the Gross–Pitaevskii equation with rotation and we prove uniform $L^{infty }$-bounds for the corresponding numerical approximations in $2D$ and $3D$ without coupling conditions between the spatial mesh size and the time step size. With this result at hand, we are particularly able to extend the previous error estimates to the $3D$ setting while avoiding artificial CFL conditions.
{"title":"Uniform L∞-bounds for energy-conserving higher-order time integrators for the Gross–Pitaevskii equation with rotation","authors":"Christian Döding, Patrick Henning","doi":"10.1093/imanum/drad081","DOIUrl":"https://doi.org/10.1093/imanum/drad081","url":null,"abstract":"In this paper, we consider an energy-conserving continuous Galerkin discretization of the Gross–Pitaevskii equation with a magnetic trapping potential and a stirring potential for angular momentum rotation. The discretization is based on finite elements in space and time and allows for arbitrary polynomial orders. It was first analyzed by O. Karakashian and C. Makridakis (SIAM J. Numer. Anal., 36(6),1779–1807, 1999) in the absence of potential terms and corresponding a priori error estimates were derived in $2D$. In this work we revisit the approach in the generalized setting of the Gross–Pitaevskii equation with rotation and we prove uniform $L^{infty }$-bounds for the corresponding numerical approximations in $2D$ and $3D$ without coupling conditions between the spatial mesh size and the time step size. With this result at hand, we are particularly able to extend the previous error estimates to the $3D$ setting while avoiding artificial CFL conditions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71524671","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}
As a specific type of shape gradient descent algorithm, shape gradient flow is widely used for shape optimization problems constrained by partial differential equations. In this approach, the constraint partial differential equations could be solved by finite element methods on a domain with a solution-driven evolving boundary. Rigorous analysis for the stability and convergence of such finite element approximations is still missing from the literature due to the complex nonlinear dependence of the boundary evolution on the solution. In this article, rigorous analysis of numerical approximations to the evolution of the boundary in a prototypical shape gradient flow is addressed. First-order convergence in time and $k$th order convergence in space for finite elements of degree $kgeqslant 2$ are proved for a linearly semi-implicit evolving finite element algorithm up to a given time. The theoretical analysis is consistent with the numerical experiments, which also illustrate the effectiveness of the proposed method in simulating two- and three-dimensional boundary evolution under shape gradient flow. The extension of the formulation, algorithm and analysis to more general shape density functions and constraint partial differential equations is also discussed.
{"title":"Convergent evolving finite element approximations of boundary evolution under shape gradient flow","authors":"Wei Gong, Buyang Li, Qiqi Rao","doi":"10.1093/imanum/drad080","DOIUrl":"https://doi.org/10.1093/imanum/drad080","url":null,"abstract":"As a specific type of shape gradient descent algorithm, shape gradient flow is widely used for shape optimization problems constrained by partial differential equations. In this approach, the constraint partial differential equations could be solved by finite element methods on a domain with a solution-driven evolving boundary. Rigorous analysis for the stability and convergence of such finite element approximations is still missing from the literature due to the complex nonlinear dependence of the boundary evolution on the solution. In this article, rigorous analysis of numerical approximations to the evolution of the boundary in a prototypical shape gradient flow is addressed. First-order convergence in time and $k$th order convergence in space for finite elements of degree $kgeqslant 2$ are proved for a linearly semi-implicit evolving finite element algorithm up to a given time. The theoretical analysis is consistent with the numerical experiments, which also illustrate the effectiveness of the proposed method in simulating two- and three-dimensional boundary evolution under shape gradient flow. The extension of the formulation, algorithm and analysis to more general shape density functions and constraint partial differential equations is also discussed.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509733","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}
Abstract We consider evolutionary systems, i.e., systems of linear partial differential equations arising from the mathematical physics. For these systems, there exists a general solution theory in exponentially weighted spaces, which can be exploited in the analysis of numerical methods. The numerical method considered in this paper is a discontinuous Galerkin method in time combined with a conforming Galerkin method in space. Building on our recent paper (Franz, S., Trostorff, S. & Waurick, M. (2019) Numerical methods for changing type systems. IMAJNA, 39, 1009–1038), we improve some of the results, study the dependence of the numerical solution on the weight parameter and consider a reformulation and post-processing of its numerical solution. As a by-product, we provide error estimates for the dG-C0 method. Numerical simulations support the theoretical findings.
{"title":"Post-processing and improved error estimates of numerical methods for evolutionary systems","authors":"Sebastian Franz","doi":"10.1093/imanum/drad082","DOIUrl":"https://doi.org/10.1093/imanum/drad082","url":null,"abstract":"Abstract We consider evolutionary systems, i.e., systems of linear partial differential equations arising from the mathematical physics. For these systems, there exists a general solution theory in exponentially weighted spaces, which can be exploited in the analysis of numerical methods. The numerical method considered in this paper is a discontinuous Galerkin method in time combined with a conforming Galerkin method in space. Building on our recent paper (Franz, S., Trostorff, S. & Waurick, M. (2019) Numerical methods for changing type systems. IMAJNA, 39, 1009–1038), we improve some of the results, study the dependence of the numerical solution on the weight parameter and consider a reformulation and post-processing of its numerical solution. As a by-product, we provide error estimates for the dG-C0 method. Numerical simulations support the theoretical findings.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134907941","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 propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations, which resemble those in Cockburn & Gopalakrishnan (2008, Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput., 74, 1653–1677) for elliptic problems, between the approximates that are obtained by the single-face hybridizable, hybrid Raviart–Thomas and hybrid Brezzi–Douglas–Marini methods. Numerical experiments underline our analytical findings.
{"title":"Homogeneous multigrid for HDG applied to the Stokes equation","authors":"Peipei Lu, Wei Wang, Guido Kanschat, Andreas Rupp","doi":"10.1093/imanum/drad079","DOIUrl":"https://doi.org/10.1093/imanum/drad079","url":null,"abstract":"We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations, which resemble those in Cockburn & Gopalakrishnan (2008, Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput., 74, 1653–1677) for elliptic problems, between the approximates that are obtained by the single-face hybridizable, hybrid Raviart–Thomas and hybrid Brezzi–Douglas–Marini methods. Numerical experiments underline our analytical findings.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516709","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 a higher-order adaptive method for strong approximations of exit times of Itô stochastic differential equations (SDEs). The method employs a strong Itô–Taylor scheme for simulating SDE paths, and adaptively decreases the step size in the numerical integration as the solution approaches the boundary of the domain. These techniques complement each other nicely: adaptive timestepping improves the accuracy of the exit time by reducing the magnitude of the overshoot of the numerical solution when it exits the domain, and higher-order schemes improve the approximation of the state of the diffusion process. We present two versions of the higher-order adaptive method. The first one uses the Milstein scheme as the numerical integrator and two step sizes for adaptive timestepping: $h$ when far away from the boundary and $h^2$ when close to the boundary. The second method is an extension of the first one using the strong Itô–Taylor scheme of order 1.5 as the numerical integrator and three step sizes for adaptive timestepping. Under some regularity assumptions, we show that for any $xi>0$, the strong error is ${mathcal{O}}(h^{1-xi })$ and ${mathcal{O}}(h^{3/2-xi })$ for the first and second method, respectively. Provided quite restrictive commutativity conditions hold for the diffusion coefficient, we further show that the expected computational cost for both methods is ${mathcal{O}}(h^{-1} log (h^{-1}))$. This results in a near doubling/trebling of the strong error rate compared to the standard Euler–Maruyama-based approach, while the computational cost rate is kept close to order one. Numerical examples that support the theoretical results are provided, and we discuss the potential for extensions that would further improve the strong convergence rate of the method.
{"title":"Higher-order adaptive methods for exit times of Itô diffusions","authors":"Håkon Hoel, Sankarasubramanian Ragunathan","doi":"10.1093/imanum/drad077","DOIUrl":"https://doi.org/10.1093/imanum/drad077","url":null,"abstract":"We construct a higher-order adaptive method for strong approximations of exit times of Itô stochastic differential equations (SDEs). The method employs a strong Itô–Taylor scheme for simulating SDE paths, and adaptively decreases the step size in the numerical integration as the solution approaches the boundary of the domain. These techniques complement each other nicely: adaptive timestepping improves the accuracy of the exit time by reducing the magnitude of the overshoot of the numerical solution when it exits the domain, and higher-order schemes improve the approximation of the state of the diffusion process. We present two versions of the higher-order adaptive method. The first one uses the Milstein scheme as the numerical integrator and two step sizes for adaptive timestepping: $h$ when far away from the boundary and $h^2$ when close to the boundary. The second method is an extension of the first one using the strong Itô–Taylor scheme of order 1.5 as the numerical integrator and three step sizes for adaptive timestepping. Under some regularity assumptions, we show that for any $xi>0$, the strong error is ${mathcal{O}}(h^{1-xi })$ and ${mathcal{O}}(h^{3/2-xi })$ for the first and second method, respectively. Provided quite restrictive commutativity conditions hold for the diffusion coefficient, we further show that the expected computational cost for both methods is ${mathcal{O}}(h^{-1} log (h^{-1}))$. This results in a near doubling/trebling of the strong error rate compared to the standard Euler–Maruyama-based approach, while the computational cost rate is kept close to order one. Numerical examples that support the theoretical results are provided, and we discuss the potential for extensions that would further improve the strong convergence rate of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509777","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 proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is unconditionally stable and that it achieves optimal convergence in $L^2$ norm for both the solution and the auxiliary variables without imposing penalty terms. A key part of the proof of the stability and convergence analysis is the special choice of the test function for the auxiliary equation involving the time derivative of the displacement variable, which leads to a linear system for the time evolution of the unknowns. Then we can use standard mathematical techniques in discontinuous Galerkin methods to obtain stability and optimal error estimates. We also obtain energy dissipation and/or conservation of the scheme by choosing simple and mesh-independent interelement fluxes. Several numerical experiments are presented to illustrate and support our theoretical results.
{"title":"A local energy-based discontinuous Galerkin method for fourth-order semilinear wave equations","authors":"Lu Zhang","doi":"10.1093/imanum/drad076","DOIUrl":"https://doi.org/10.1093/imanum/drad076","url":null,"abstract":"This paper proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is unconditionally stable and that it achieves optimal convergence in $L^2$ norm for both the solution and the auxiliary variables without imposing penalty terms. A key part of the proof of the stability and convergence analysis is the special choice of the test function for the auxiliary equation involving the time derivative of the displacement variable, which leads to a linear system for the time evolution of the unknowns. Then we can use standard mathematical techniques in discontinuous Galerkin methods to obtain stability and optimal error estimates. We also obtain energy dissipation and/or conservation of the scheme by choosing simple and mesh-independent interelement fluxes. Several numerical experiments are presented to illustrate and support our theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509773","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}
Franco Dassi, Alessio Fumagalli, Ilario Mazzieri, Giuseppe Vacca
We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In the absence of external loads, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical $theta $-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing an optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering applications of the method.
{"title":"Mixed Virtual Element approximation of linear acoustic wave equation","authors":"Franco Dassi, Alessio Fumagalli, Ilario Mazzieri, Giuseppe Vacca","doi":"10.1093/imanum/drad078","DOIUrl":"https://doi.org/10.1093/imanum/drad078","url":null,"abstract":"We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In the absence of external loads, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical $theta $-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing an optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering applications of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509812","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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