Daniel Eckhardt, Marlis Hochbruck, Barbara Verfürth
We present an implicit–explicit (IMEX) scheme for semilinear wave equations with strong damping. By treating the nonlinear, nonstiff term explicitly and the linear, stiff part implicitly we obtain a method that is, not only unconditionally stable, but also highly efficient. Our main results are error bounds of the full discretization in space and time for the IMEX scheme combined with a general abstract space discretization. As an application we consider the heterogeneous multiscale method for wave equations with highly oscillating coefficients in space for which we show spatial and temporal convergence rates by using the abstract result.
{"title":"Error analysis of an implicit–explicit time discretization scheme for semilinear wave equations with application to multiscale problems","authors":"Daniel Eckhardt, Marlis Hochbruck, Barbara Verfürth","doi":"10.1093/imanum/draf092","DOIUrl":"https://doi.org/10.1093/imanum/draf092","url":null,"abstract":"We present an implicit–explicit (IMEX) scheme for semilinear wave equations with strong damping. By treating the nonlinear, nonstiff term explicitly and the linear, stiff part implicitly we obtain a method that is, not only unconditionally stable, but also highly efficient. Our main results are error bounds of the full discretization in space and time for the IMEX scheme combined with a general abstract space discretization. As an application we consider the heterogeneous multiscale method for wave equations with highly oscillating coefficients in space for which we show spatial and temporal convergence rates by using the abstract result.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145295940","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 the nonrelativistic limit regime, nonlinear Dirac equations involve a small parameter $varepsilon>0$ which induces rapid temporal oscillations with frequency proportional to $varepsilon ^{-2}$. Efficient time integrators are challenging to construct, since their accuracy has to be independent of $varepsilon $ or improve with smaller values of $varepsilon $. Yongyong Cai and Yan Wang have presented a nested Picard iterative integrator (NPI-2), which is a uniformly accurate second-order scheme. We propose a novel method called the nonresonant nested Picard iterative integrator (NRNPI), which takes advantage of cancelation effects in the global error to significantly simplify the NPI-2. We prove that for nonresonant step sizes $tau geq frac{pi }{4} varepsilon ^{2}$, the NRNPI has the same accuracy as the NPI-2 and is thus more efficient. Moreover, we show that for arbitrary $tau < frac{pi }{4} varepsilon ^{2}$, the error decreases proportionally to $varepsilon ^{2} tau $. We provide numerical experiments to illustrate the error behavior as well as the efficiency gain.
{"title":"Employing nonresonant step sizes for time integration of highly oscillatory nonlinear Dirac equations","authors":"Tobias Jahnke, Michael Kirn","doi":"10.1093/imanum/draf085","DOIUrl":"https://doi.org/10.1093/imanum/draf085","url":null,"abstract":"In the nonrelativistic limit regime, nonlinear Dirac equations involve a small parameter $varepsilon&gt;0$ which induces rapid temporal oscillations with frequency proportional to $varepsilon ^{-2}$. Efficient time integrators are challenging to construct, since their accuracy has to be independent of $varepsilon $ or improve with smaller values of $varepsilon $. Yongyong Cai and Yan Wang have presented a nested Picard iterative integrator (NPI-2), which is a uniformly accurate second-order scheme. We propose a novel method called the nonresonant nested Picard iterative integrator (NRNPI), which takes advantage of cancelation effects in the global error to significantly simplify the NPI-2. We prove that for nonresonant step sizes $tau geq frac{pi }{4} varepsilon ^{2}$, the NRNPI has the same accuracy as the NPI-2 and is thus more efficient. Moreover, we show that for arbitrary $tau &lt; frac{pi }{4} varepsilon ^{2}$, the error decreases proportionally to $varepsilon ^{2} tau $. We provide numerical experiments to illustrate the error behavior as well as the efficiency gain.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145247025","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 prove that the spatially semi-discrete evolving finite element methods (FEMs) for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^{p}$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving FEM, properties of Green’s functions on (discretized) closed surfaces, and local energy estimates for FEMs.
{"title":"Maximal regularity of evolving FEMs for parabolic equations on an evolving surface","authors":"Genming Bai, Balázs Kovács, Buyang Li","doi":"10.1093/imanum/draf082","DOIUrl":"https://doi.org/10.1093/imanum/draf082","url":null,"abstract":"In this paper, we prove that the spatially semi-discrete evolving finite element methods (FEMs) for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^{p}$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving FEM, properties of Green’s functions on (discretized) closed surfaces, and local energy estimates for FEMs.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"39 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145209782","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 is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF) can be found. A central constituent is the notion of variationally correct residual loss function, meaning that its value is always uniformly proportional to the squared solution error in the norm determined by the SVF, hence facilitating rigorous a posteriori accuracy control. It is based on a single variational problem, associated with the family of parameter-dependent fibre problems, employing the notion of direct integrals of Hilbert spaces. Since in its original form the loss function is given as a dual test norm of the residual; a central objective is to develop equivalent computable expressions. The first critical role is played by hybrid hypothesis classes, whose elements are piecewise polynomial in (low-dimensional) spatio-temporal variables with parameter-dependent coefficients that can be represented, for example, by neural networks. Second, working with first-order SVFs we distinguish two scenarios: (i) the test space can be chosen as an $L_{2}$-space (such as for elliptic or parabolic problems) so that residuals can be evaluated directly as elements of $L_{2}$; (ii) when trial and test spaces for the fibre problems depend on the parameters (as for transport equations) we use ultra-weak formulations. In combination with discontinuous Petrov–Galerkin concepts the hybrid format is then instrumental to arrive at variationally correct computable residual loss functions. Our findings are illustrated by numerical experiments representing (i) and (ii), namely elliptic boundary value problems with piecewise constant diffusion coefficients and pure transport equations with parameter-dependent convection fields.
{"title":"Variationally correct neural residual regression for parametric PDEs: on the viability of controlled accuracy","authors":"Markus Bachmayr, Wolfgang Dahmen, Mathias Oster","doi":"10.1093/imanum/draf073","DOIUrl":"https://doi.org/10.1093/imanum/draf073","url":null,"abstract":"This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF) can be found. A central constituent is the notion of variationally correct residual loss function, meaning that its value is always uniformly proportional to the squared solution error in the norm determined by the SVF, hence facilitating rigorous a posteriori accuracy control. It is based on a single variational problem, associated with the family of parameter-dependent fibre problems, employing the notion of direct integrals of Hilbert spaces. Since in its original form the loss function is given as a dual test norm of the residual; a central objective is to develop equivalent computable expressions. The first critical role is played by hybrid hypothesis classes, whose elements are piecewise polynomial in (low-dimensional) spatio-temporal variables with parameter-dependent coefficients that can be represented, for example, by neural networks. Second, working with first-order SVFs we distinguish two scenarios: (i) the test space can be chosen as an $L_{2}$-space (such as for elliptic or parabolic problems) so that residuals can be evaluated directly as elements of $L_{2}$; (ii) when trial and test spaces for the fibre problems depend on the parameters (as for transport equations) we use ultra-weak formulations. In combination with discontinuous Petrov–Galerkin concepts the hybrid format is then instrumental to arrive at variationally correct computable residual loss functions. Our findings are illustrated by numerical experiments representing (i) and (ii), namely elliptic boundary value problems with piecewise constant diffusion coefficients and pure transport equations with parameter-dependent convection fields.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145209783","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 and analyze a space–time finite element method for Westervelt’s quasilinear model of ultrasound waves in its second-order formulation. The method combines conforming finite element spatial discretizations with a discontinuous–continuous Galerkin time stepping. Its analysis is challenged by the fact that standard Galerkin testing approaches for wave problems do not allow for bounding the discrete energy at all times. By means of redesigned energy arguments for a linearized problem combined with Banach’s fixed-point argument, we show the well-posedness of the scheme, a priori error estimates, and robustness with respect to the strong damping parameter $delta $. Moreover, the scheme preserves the asymptotic preserving property of the continuous problem; more precisely, we prove that the discrete solutions corresponding to $delta>0$ converge, in the singular vanishing dissipation limit, to the solution of the discrete inviscid problem. We use several numerical experiments in $(2 + 1)$ dimensions to validate our theoretical results.
{"title":"Combined DG–CG finite element method for the Westervelt equation","authors":"Sergio Gómez, Vanja Nikolić","doi":"10.1093/imanum/draf080","DOIUrl":"https://doi.org/10.1093/imanum/draf080","url":null,"abstract":"We propose and analyze a space–time finite element method for Westervelt’s quasilinear model of ultrasound waves in its second-order formulation. The method combines conforming finite element spatial discretizations with a discontinuous–continuous Galerkin time stepping. Its analysis is challenged by the fact that standard Galerkin testing approaches for wave problems do not allow for bounding the discrete energy at all times. By means of redesigned energy arguments for a linearized problem combined with Banach’s fixed-point argument, we show the well-posedness of the scheme, a priori error estimates, and robustness with respect to the strong damping parameter $delta $. Moreover, the scheme preserves the asymptotic preserving property of the continuous problem; more precisely, we prove that the discrete solutions corresponding to $delta&gt;0$ converge, in the singular vanishing dissipation limit, to the solution of the discrete inviscid problem. We use several numerical experiments in $(2 + 1)$ dimensions to validate our theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145183021","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 analyse the preservation of asymptotic properties of partially dissipative hyperbolic systems when switching to a fully discrete setting. We prove that one of the simplest consistent and unconditionally stable numerical methods—the implicit central finite-difference scheme—preserves both the large time asymptotic behaviour and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems that satisfy the Kalman rank condition. The large time asymptotic-preserving property is achieved by conceiving time-weighted perturbed energy functionals in the spirit of the hypocoercivity theory. For the relaxation-preserving property, drawing inspiration from the observation that, in the continuous case, solutions are shown to exhibit distinct behaviour in low and high frequencies we introduce a novel discrete Littlewood–Paley decomposition tailored to the central finite-difference scheme. This allows us to prove Bernstein-type estimates for discrete differential operators and leads to new diffusive limit results such as the strong convergence of the discrete linearized compressible Euler system with damping towards the discrete heat equation, uniformly with respect to the spatial mesh parameter.
{"title":"Asymptotic-preserving finite difference method for partially dissipative hyperbolic systems","authors":"Timothée Crin-Barat, Dragoș Manea","doi":"10.1093/imanum/draf066","DOIUrl":"https://doi.org/10.1093/imanum/draf066","url":null,"abstract":"We analyse the preservation of asymptotic properties of partially dissipative hyperbolic systems when switching to a fully discrete setting. We prove that one of the simplest consistent and unconditionally stable numerical methods—the implicit central finite-difference scheme—preserves both the large time asymptotic behaviour and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems that satisfy the Kalman rank condition. The large time asymptotic-preserving property is achieved by conceiving time-weighted perturbed energy functionals in the spirit of the hypocoercivity theory. For the relaxation-preserving property, drawing inspiration from the observation that, in the continuous case, solutions are shown to exhibit distinct behaviour in low and high frequencies we introduce a novel discrete Littlewood–Paley decomposition tailored to the central finite-difference scheme. This allows us to prove Bernstein-type estimates for discrete differential operators and leads to new diffusive limit results such as the strong convergence of the discrete linearized compressible Euler system with damping towards the discrete heat equation, uniformly with respect to the spatial mesh parameter.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145133442","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}
Based on the proximal bundle and gradient sampling (GS) methods, we develop a robust algorithm for minimizing the locally Lipschitz function $f:mathbb{R}^{n}to mathbb{R}$. As an interesting feature of the proposed method, thanks to the GS technique, we sample a set of differentiable auxiliary points from the vicinity of the current point to construct an initial piecewise linear model for the objective function. If necessary, inspired by bundle methods, we iteratively enrich the set of sampled points by using a single nonredundant auxiliary point suggested by a modified variant of Mifflin’s line search. However, we may terminate the enrichment process without achieving a descent step, which is different from classic bundle methods. Indeed, the proposed enrichment process only accepts those auxiliary points having a small gradient locality measure, which significantly improves the efficiency of the method in practice. In theory, our method keeps iterations where the objective function is differentiable, and consequently, it works only with the gradient vectors of the objective function. In contrast with existing GS methods, the radius of the sampling region is not monotone. More precisely, by proposing a nonmonotone proximity parameter based on the radius of the sampling region, we add some valuable features of the trust region philosophy to our algorithm. The convergence analysis of the proposed method is comprehensively studied using exact and inexact gradients. By means of various academic and semi-academic test problems, we demonstrate the reliability and efficiency of the proposed method in practice.1
{"title":"A bundle-trust method via gradient sampling technique for nonsmooth optimization using exact and inexact gradients","authors":"Morteza Maleknia, Majid Soleimani-damaneh","doi":"10.1093/imanum/draf087","DOIUrl":"https://doi.org/10.1093/imanum/draf087","url":null,"abstract":"Based on the proximal bundle and gradient sampling (GS) methods, we develop a robust algorithm for minimizing the locally Lipschitz function $f:mathbb{R}^{n}to mathbb{R}$. As an interesting feature of the proposed method, thanks to the GS technique, we sample a set of differentiable auxiliary points from the vicinity of the current point to construct an initial piecewise linear model for the objective function. If necessary, inspired by bundle methods, we iteratively enrich the set of sampled points by using a single nonredundant auxiliary point suggested by a modified variant of Mifflin’s line search. However, we may terminate the enrichment process without achieving a descent step, which is different from classic bundle methods. Indeed, the proposed enrichment process only accepts those auxiliary points having a small gradient locality measure, which significantly improves the efficiency of the method in practice. In theory, our method keeps iterations where the objective function is differentiable, and consequently, it works only with the gradient vectors of the objective function. In contrast with existing GS methods, the radius of the sampling region is not monotone. More precisely, by proposing a nonmonotone proximity parameter based on the radius of the sampling region, we add some valuable features of the trust region philosophy to our algorithm. The convergence analysis of the proposed method is comprehensively studied using exact and inexact gradients. By means of various academic and semi-academic test problems, we demonstrate the reliability and efficiency of the proposed method in practice.1","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145093586","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 purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one-parameter family of cost functions that interpolates between the original and a special cost function for which the problem’s complexity scales linearly in $m$. We then show that the solution to the original problem can be recovered by solving an ordinary differential equation in the parameter $varepsilon $, whose initial condition corresponds to the solution for the special cost function mentioned above; we then present some simulations, using both explicit Euler and explicit higher order Runge–Kutta schemes to compute solutions to the ordinary differential equation, and, as a result, the multi-marginal optimal transport problem.
{"title":"An ODE characterization of multi-marginal optimal transport with pairwise cost functions","authors":"Luca Nenna, Brendan Pass","doi":"10.1093/imanum/draf067","DOIUrl":"https://doi.org/10.1093/imanum/draf067","url":null,"abstract":"The purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one-parameter family of cost functions that interpolates between the original and a special cost function for which the problem’s complexity scales linearly in $m$. We then show that the solution to the original problem can be recovered by solving an ordinary differential equation in the parameter $varepsilon $, whose initial condition corresponds to the solution for the special cost function mentioned above; we then present some simulations, using both explicit Euler and explicit higher order Runge–Kutta schemes to compute solutions to the ordinary differential equation, and, as a result, the multi-marginal optimal transport problem.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"64 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003090","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}
Forward-backward splitting (FBS) is one of the most fundamental and efficient optimization algorithms in linear inverse problems like sparse recovery and image reconstruction, and has recently been unrolled to construct several deep neural networks with dramatic performance advantages over conventional methods. This circumstance motivates us to consider some theoretical aspects of the basic FBS-induced network. Here, ‘basic’ means that the neural network is unrolled from the original FBS algorithm with direct parameter relaxation. In this paper we report the first part of our study, i.e., deep-layer limit behavior and stability of feed-forward systems. We formulate the finite layer network as a difference inclusion and model the related deep-layer limit system as a differential inclusion. We mainly analyze the uniform convergence properties from the state of the finite layer network to that of the related deep-layer limit system, as well as their forward stability. Our analysis procedure can be simplified to analyze the LISTA- and ALISTA-like networks. A numerical example is implemented to illustrate the convergence results and perturbation stability. As a side product of this study, some corollaries in the case of pointwise sampling and Lipschitz continuity assumptions provide convergence results in the context of numerical ordinary differential inclusion.
{"title":"Deep-layer limit and stability analysis of the basic forward–backward-splitting induced network (I): feed-forward systems","authors":"Xuan Lin, Chunlin Wu","doi":"10.1093/imanum/draf068","DOIUrl":"https://doi.org/10.1093/imanum/draf068","url":null,"abstract":"Forward-backward splitting (FBS) is one of the most fundamental and efficient optimization algorithms in linear inverse problems like sparse recovery and image reconstruction, and has recently been unrolled to construct several deep neural networks with dramatic performance advantages over conventional methods. This circumstance motivates us to consider some theoretical aspects of the basic FBS-induced network. Here, ‘basic’ means that the neural network is unrolled from the original FBS algorithm with direct parameter relaxation. In this paper we report the first part of our study, i.e., deep-layer limit behavior and stability of feed-forward systems. We formulate the finite layer network as a difference inclusion and model the related deep-layer limit system as a differential inclusion. We mainly analyze the uniform convergence properties from the state of the finite layer network to that of the related deep-layer limit system, as well as their forward stability. Our analysis procedure can be simplified to analyze the LISTA- and ALISTA-like networks. A numerical example is implemented to illustrate the convergence results and perturbation stability. As a side product of this study, some corollaries in the case of pointwise sampling and Lipschitz continuity assumptions provide convergence results in the context of numerical ordinary differential inclusion.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"130 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144919399","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 propose and analyze a mixed formulation for the Kelvin–Voigt–Brinkman–Forchheimer equations for unsteady viscoelastic flows in porous media. Besides the velocity and pressure, our approach introduces the vorticity as a further unknown. Consequently, we obtain a three-field mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish the existence and uniqueness of a solution for the weak formulation, and derive the corresponding stability bounds, employing a fixed-point strategy, along with monotone operators theory and Schauder theorem. Afterwards, we introduce a semidiscrete continuous-in-time approximation based on stable Stokes elements for the velocity and pressure, and continuous or discontinuous piecewise polynomial spaces for the vorticity. Additionally, employing backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness, derive stability bounds and establish the corresponding error estimates for both schemes. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.
{"title":"Velocity-vorticity-pressure mixed formulation for the Kelvin–Voigt–Brinkman–Forchheimer equations","authors":"Sergio Caucao, Ivan Yotov","doi":"10.1093/imanum/draf072","DOIUrl":"https://doi.org/10.1093/imanum/draf072","url":null,"abstract":"In this paper, we propose and analyze a mixed formulation for the Kelvin–Voigt–Brinkman–Forchheimer equations for unsteady viscoelastic flows in porous media. Besides the velocity and pressure, our approach introduces the vorticity as a further unknown. Consequently, we obtain a three-field mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish the existence and uniqueness of a solution for the weak formulation, and derive the corresponding stability bounds, employing a fixed-point strategy, along with monotone operators theory and Schauder theorem. Afterwards, we introduce a semidiscrete continuous-in-time approximation based on stable Stokes elements for the velocity and pressure, and continuous or discontinuous piecewise polynomial spaces for the vorticity. Additionally, employing backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness, derive stability bounds and establish the corresponding error estimates for both schemes. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"38 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144919237","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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