IMA Journal of Numerical Analysis最新文献

{"title":"Discrete anisotropic curve shortening flow in higher codimension","authors":"Klaus Deckelnick, Robert Nürnberg","doi":"10.1093/imanum/drae015","DOIUrl":"https://doi.org/10.1093/imanum/drae015","url":null,"abstract":"We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{mathbb{R}}}^{d}$, $dgeq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"73 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096758","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}

{"title":"High-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation","authors":"Rodolfo Bermejo, Manuel Colera","doi":"10.1093/imanum/drae018","DOIUrl":"https://doi.org/10.1093/imanum/drae018","url":null,"abstract":"We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $mu $ is large, the error is $O(h^{k+1}+varDelta t^{q})$, whereas in the advective regime, $mu ll 1$, the convergence is $O(min (h^{k},frac{h^{k+1} }{varDelta t})+varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $mu ^{-1}$ dependence.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"127 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954632","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}

{"title":"An equilibrated estimator for mixed finite element discretizations of the curl-curl problem","authors":"T Chaumont-Frelet","doi":"10.1093/imanum/drae007","DOIUrl":"https://doi.org/10.1093/imanum/drae007","url":null,"abstract":"We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"126 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954005","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}

{"title":"Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations","authors":"Xiaojing Dong, Huayi Huang, Yunqing Huang, Xiaojuan Shen, Qili Tang","doi":"10.1093/imanum/drae016","DOIUrl":"https://doi.org/10.1093/imanum/drae016","url":null,"abstract":"In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $beta =0$ and RPC-SAV-CN when $beta in (0,frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-alpha _{1}varDelta tnu varDelta (widetilde {textbf {u}}^{n+1}-{textbf {u}}^{n})$ and $alpha _{2}varDelta tsigma ^{-1}mbox {curl}mbox {curl} (textbf {H}^{n+1}-textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"48 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140949377","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}

{"title":"Efficient function approximation in enriched approximation spaces","authors":"Astrid Herremans, Daan Huybrechs","doi":"10.1093/imanum/drae017","DOIUrl":"https://doi.org/10.1093/imanum/drae017","url":null,"abstract":"An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can still provide highly accurate numerical approximations under reasonable conditions. In this paper we propose an efficient algorithm to compute such approximations. It is based on the AZ algorithm for overcomplete sets and frames, which simplifies in the case of an enriched basis. In addition, analysis of the original AZ algorithm and of the proposed variant gives constructive insights on how to achieve optimal and stable discretizations using enriched bases. We apply the algorithm to examples of enriched approximation spaces in literature, including a few nonstandard approximation problems and an enriched spectral method for a 2D boundary value problem, and show that the simplified AZ algorithm is indeed stable, accurate and efficient.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140915068","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}

{"title":"Variational data assimilation with finite-element discretization for second-order parabolic interface equation","authors":"Xuejian Li, Xiaoming He, Wei Gong, Craig C Douglas","doi":"10.1093/imanum/drae010","DOIUrl":"https://doi.org/10.1093/imanum/drae010","url":null,"abstract":"In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140910594","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}

{"title":"Error analysis for local discontinuous Galerkin semidiscretization of Richards’ equation","authors":"Scott Congreve, Vít Dolejší, Sunčica Sakić","doi":"10.1093/imanum/drae013","DOIUrl":"https://doi.org/10.1093/imanum/drae013","url":null,"abstract":"This paper concerns an error analysis of the space semidiscrete scheme for the Richards’ equation modeling flows in variably saturated porous media. This nonlinear parabolic partial differential equation can degenerate; namely, we consider the case where the time derivative term can vanish, i.e., the fast-diffusion type of degeneracy. We discretize the Richards’ equation by the local discontinuous Galerkin method, which provides high order accuracy and preserves stability. Due to the nonlinearity of the problem, special techniques for numerical analysis of the scheme are required. In particular, we combine two partial error bounds using continuous mathematical induction and derive a priori error estimates with respect to the spatial discretization parameter and the Hölder coefficient of the nonlinear temporal derivative. Finally, the theoretical results are supported by numerical experiments, including cases beyond the assumptions of the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"44 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140910600","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}

{"title":"A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients","authors":"Mengchao Wang, Xiaojie Wang","doi":"10.1093/imanum/drae012","DOIUrl":"https://doi.org/10.1093/imanum/drae012","url":null,"abstract":"The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ mathcal{D} in{mathbb{R}}^{d}$, $ d leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($text{Tr} (Q) < infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"50 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140895675","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}

{"title":"Stability of convergence rates: kernel interpolation on non-Lipschitz domains","authors":"Tizian Wenzel, Gabriele Santin, Bernard Haasdonk","doi":"10.1093/imanum/drae014","DOIUrl":"https://doi.org/10.1093/imanum/drae014","url":null,"abstract":"Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $varOmega subset{mathbb{R}} ^{d}$, thus allowing for non-Lipschitz domains including e.g., cusps and irregular boundaries. Especially we show that, when going to a smaller domain $tilde{varOmega } subset varOmega subset{mathbb{R}} ^{d}$, the convergence rate does not deteriorate—i.e., the convergence rates are stable with respect to going to a subset. We obtain this by leveraging an analysis of greedy kernel algorithms. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $varOmega $ has an impact on the approximation properties. Numerical experiments illustrate and confirm the analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140895686","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}

{"title":"Strong convergence of adaptive time-stepping schemes for the stochastic Allen–Cahn equation","authors":"Chuchu Chen, Tonghe Dang, Jialin Hong","doi":"10.1093/imanum/drae009","DOIUrl":"https://doi.org/10.1093/imanum/drae009","url":null,"abstract":"It is known from Beccari et al. (2019) that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen–Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration to control numerical solutions from instability. The a priori estimates in $mathscr{C}(mathscr{O})$-norm and $dot{H}^{beta }(mathscr{O})$-norm of numerical solutions are established provided the adaptive timestep function is suitably bounded, which plays a key role in the convergence analysis. We show that the adaptive time-stepping schemes converge strongly with order $frac{beta }{2}$ in time and $frac{beta }{d}$ in space with $d$ ($d=1,2,3$) being the dimension and $beta in (0,2]$. Numerical experiments show that the adaptive time-stepping schemes are simple to implement and at a lower computational cost than a scheme with the uniform timestep.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826360","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}