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A-posteriori error estimates for systems of hyperbolic conservation laws via computing negative norms of local residuals
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-11 DOI: 10.1093/imanum/drae111
Jan Giesselmann, Aleksey Sikstel
{"title":"A-posteriori error estimates for systems of hyperbolic conservation laws via computing negative norms of local residuals","authors":"Jan Giesselmann, Aleksey Sikstel","doi":"10.1093/imanum/drae111","DOIUrl":"https://doi.org/10.1093/imanum/drae111","url":null,"abstract":"We prove rigorous a-posteriori error estimates for first-order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by Bressan, Chiri and Shen, a new way to localize residuals and a novel method to compute negative-order norms of these local residuals. Computing negative-order norms becomes possible by suitably projecting test functions onto a finite dimensional space. Numerical experiments show that the error estimator converges with the rate predicted by a-priori error estimates.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599894","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}
引用次数: 0
Well-posedness of first-order acoustic wave equations and space-time finite element approximation 一阶声波方程的良好拟合与时空有限元近似
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-11 DOI: 10.1093/imanum/drae104
Thomas Führer, Roberto González, Michael Karkulik
{"title":"Well-posedness of first-order acoustic wave equations and space-time finite element approximation","authors":"Thomas Führer, Roberto González, Michael Karkulik","doi":"10.1093/imanum/drae104","DOIUrl":"https://doi.org/10.1093/imanum/drae104","url":null,"abstract":"We study a first-order system formulation of the (acoustic) wave equation and prove that the operator of this system is an isomorphism from an appropriately defined graph space to $L^{2}$. The results rely on well-posedness and stability of the weak and ultraweak formulation of the second-order wave equation. As an application, we define and analyze a space-time least-squares finite element method for solving the wave equation. Some numerical examples for one- and two-dimensional spatial domains are presented.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"86 1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599893","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}
引用次数: 0
A noncoforming virtual element approximation for the Oseen eigenvalue problem
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-11 DOI: 10.1093/imanum/drae108
Dibyendu Adak, Felipe Lepe, Gonzalo Rivera
{"title":"A noncoforming virtual element approximation for the Oseen eigenvalue problem","authors":"Dibyendu Adak, Felipe Lepe, Gonzalo Rivera","doi":"10.1093/imanum/drae108","DOIUrl":"https://doi.org/10.1093/imanum/drae108","url":null,"abstract":"In this paper, we analyze a nonconforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. The spaces under consideration lead to a divergence-free method that is capable to capture properly the divergence at discrete level and the eigenvalues and eigenfunctions. Under the compact theory for operators, we prove convergence and error estimates for the method. By employing the theory of compact operators, we recover the double order of convergence of the spectrum. Finally, we present numerical tests to assess the performance of the proposed numerical scheme.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599895","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}
引用次数: 0
Geometry error analysis of a parametric mapping for higher order unfitted space–time methods 高阶非拟合时空方法参数映射的几何误差分析
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-10 DOI: 10.1093/imanum/drae098
Fabian Heimann, Christoph Lehrenfeld
{"title":"Geometry error analysis of a parametric mapping for higher order unfitted space–time methods","authors":"Fabian Heimann, Christoph Lehrenfeld","doi":"10.1093/imanum/drae098","DOIUrl":"https://doi.org/10.1093/imanum/drae098","url":null,"abstract":"In Heimann, Lehrenfeld, and Preuß (2023, SIAM J. Sci. Comp., 45(2), B139–B165), new geometrically unfitted space–time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and time have been introduced. For geometrically higher-order accuracy a parametric mapping on a background space–time tensor-product mesh has been used. In this paper, we concentrate on the geometrical accuracy of the approximation and derive rigorous bounds for the distance between the realized and an ideal mapping in different norms and derive results for the space–time regularity of the parametric mapping. These results are important and lay the ground for the error analysis of corresponding unfitted space–time finite element methods.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599897","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}
引用次数: 0
Analysis and finite element approximation of a diffuse interface approach to the Stokes–Biot coupling
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-10 DOI: 10.1093/imanum/draf002
Francis R A Aznaran, Martina Bukač, Boris Muha, Abner J Salgado
{"title":"Analysis and finite element approximation of a diffuse interface approach to the Stokes–Biot coupling","authors":"Francis R A Aznaran, Martina Bukač, Boris Muha, Abner J Salgado","doi":"10.1093/imanum/draf002","DOIUrl":"https://doi.org/10.1093/imanum/draf002","url":null,"abstract":"We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $mathcal{O}(varepsilon)$ around the interface; this allows the weak forms to be integrated uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretization of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We, in turn, prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"54 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599896","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}
引用次数: 0
Tensorized block rational Krylov methods for tensor Sylvester equations
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-05 DOI: 10.1093/imanum/draf001
Angelo A Casulli
{"title":"Tensorized block rational Krylov methods for tensor Sylvester equations","authors":"Angelo A Casulli","doi":"10.1093/imanum/draf001","DOIUrl":"https://doi.org/10.1093/imanum/draf001","url":null,"abstract":"We introduce the definition of tensorized block rational Krylov subspace and its relation with multivariate rational functions, extending the formulation of tensorized Krylov subspaces introduced in Kressner, D. & Tobler, C. (2010) Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl., 31,$1688$–$1714$. Moreover, we develop methods for the solution of tensor Sylvester equations with low multilinear or tensor train rank, based on projection onto a tensor block rational Krylov subspace. We provide a convergence analysis, some strategies for pole selection and techniques to efficiently compute the residual.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546177","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}
引用次数: 0
Scaling-robust built-in a posteriori error estimation for discontinuous least-squares finite element methods
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-05 DOI: 10.1093/imanum/drae105
Philipp Bringmann
{"title":"Scaling-robust built-in a posteriori error estimation for discontinuous least-squares finite element methods","authors":"Philipp Bringmann","doi":"10.1093/imanum/drae105","DOIUrl":"https://doi.org/10.1093/imanum/drae105","url":null,"abstract":"A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions by the example of the Poisson model problem with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables the penalization of the jump with the natural power of the mesh size in the least-squares functional. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments illustrate the importance of the proposed weighting and exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"35 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546180","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}
引用次数: 0
Complexity guarantees for nonconvex Newton-MR under inexact Hessian information
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-05 DOI: 10.1093/imanum/drae110
Alexander Lim, Fred Roosta
{"title":"Complexity guarantees for nonconvex Newton-MR under inexact Hessian information","authors":"Alexander Lim, Fred Roosta","doi":"10.1093/imanum/drae110","DOIUrl":"https://doi.org/10.1093/imanum/drae110","url":null,"abstract":"We consider an extension of the Newton-MR algorithm for nonconvex unconstrained optimization to the settings where Hessian information is approximated. Under a particular noise model on the Hessian matrix, we investigate the iteration and operation complexities of this variant to achieve appropriate sub-optimality criteria in several nonconvex settings. We do this by first considering functions that satisfy the (generalized) Polyak–Łojasiewicz condition, a special sub-class of nonconvex functions. We show that, under certain conditions, our algorithm achieves global linear convergence rate. We then consider more general nonconvex settings where the rate to obtain first-order sub-optimality is shown to be sub-linear. In all these settings we show that our algorithm converges regardless of the degree of approximation of the Hessian as well as the accuracy of the solution to the sub-problem. Finally, we compare the performance of our algorithm with several alternatives on a few machine learning problems.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"101 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546178","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}
引用次数: 0
Parametric finite element approximation of two-phase Navier–Stokes flow with viscoelasticity
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-02-24 DOI: 10.1093/imanum/drae103
Harald Garcke, Robert Nürnberg, Dennis Trautwein
{"title":"Parametric finite element approximation of two-phase Navier–Stokes flow with viscoelasticity","authors":"Harald Garcke, Robert Nürnberg, Dennis Trautwein","doi":"10.1093/imanum/drae103","DOIUrl":"https://doi.org/10.1093/imanum/drae103","url":null,"abstract":"In this work we present a parametric finite element approximation of two-phase Navier–Stokes flow with viscoelasticity. The free boundary problem is given by the viscoelastic Navier–Stokes equations in the two fluid phases, connected by jump conditions across the interface. The elasticity in the fluids is characterized using the Oldroyd-B model with possible stress diffusion. The model was originally introduced to approximate fluid-structure interaction problems between an incompressible Newtonian fluid and a hyperelastic neo-Hookean solid, which are possible limit cases of the model. We approximate a variational formulation of the model with an unfitted finite element method that uses piecewise linear parametric finite elements. The two-phase Navier–Stokes–Oldroyd-B system in the bulk regions is discretized in a way that guarantees unconditional solvability and stability for the coupled bulk–interface system. Good volume conservation properties for the two phases are observed in the case where the pressure approximation space is enriched with the help of an extended finite element method function. We show the applicability of our method with some numerical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143485893","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}
引用次数: 0
Smoothed circulant embedding with applications to multilevel Monte Carlo methods for PDEs with random coefficients
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-02-22 DOI: 10.1093/imanum/drae102
Anastasia Istratuca, Aretha L Teckentrup
{"title":"Smoothed circulant embedding with applications to multilevel Monte Carlo methods for PDEs with random coefficients","authors":"Anastasia Istratuca, Aretha L Teckentrup","doi":"10.1093/imanum/drae102","DOIUrl":"https://doi.org/10.1093/imanum/drae102","url":null,"abstract":"We consider the computational efficiency of Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to partial differential equations with random coefficients. These arise, for example, in groundwater flow modelling, where a commonly used model for the unknown parameter is a random field. We use the circulant embedding procedure for sampling from the aforementioned coefficient. To improve the computational complexity of the MLMC estimator in the case of highly oscillatory random fields we devise and implement a smoothing technique integrated into the circulant embedding method. This allows us to choose the coarsest mesh on the first level of MLMC independently of the correlation length of the covariance function of the random field, leading to considerable savings in computational cost. We illustrate this with numerical experiments, where we see a saving of up to factor 5–10 in computational cost for accuracies of practical interest.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143473557","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}
引用次数: 0
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