Advances in Computational Mathematics最新文献

{"title":"A method of fundamental solutions for large-scale 3D elastance and mobility problems","authors":"Anna Broms,&nbsp;Alex H. Barnett,&nbsp;Anna-Karin Tornberg","doi":"10.1007/s10444-025-10258-4","DOIUrl":"10.1007/s10444-025-10258-4","url":null,"abstract":"<div><p>The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here, we present new scalable MFS formulations for the corresponding elastance and mobility problems. The elastance problem computes the potentials of conductors with given net charges, while the mobility problem—crucial to rheology and complex fluid applications—computes rigid body velocities given net forces and torques on the particles. The key idea is orthogonal projection of the net charge (or forces and torques) in a rectangular variant of a “completion flow.” The proposal is compatible with one-body preconditioning, resulting in well-conditioned square linear systems amenable to fast multipole accelerated iterative solution, thus a cost linear in the particle number. For large suspensions with moderate lubrication forces, MFS sources on inner proxy-surfaces give accuracy on par with a well-resolved boundary integral formulation. Our several numerical tests include a suspension of 10,000 nearby ellipsoids, using <span>(2.6times 10^7)</span> total preconditioned degrees of freedom, where GMRES converges to five digits of accuracy in under two hours on one workstation.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 5","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10258-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An iterative projection method for unsteady Navier–Stokes equations with high Reynolds numbers","authors":"Xiaoming Zheng,&nbsp;Kun Zhao,&nbsp;Jiahong Wu,&nbsp;Weiwei Hu,&nbsp;Dapeng Du","doi":"10.1007/s10444-025-10257-5","DOIUrl":"10.1007/s10444-025-10257-5","url":null,"abstract":"<div><p>A new iterative projection method is proposed to solve the unsteady Navier–Stokes equations with high Reynolds numbers. The convectional projection method attempts to project the intermediate velocity to the divergence-free space only once per time step. However, such a velocity is not genuinely divergence-free in general practice, which can yield large errors when the Reynolds number is high. The new method has several important features: the BDF2 time discretization, the skew-symmetric convection in a semi-implicit form, two modulating parameters, and the iterative projections in each time step. A major difficulty in the proof of iteration convergence is the nonlinear convection. We solve this problem by first analyzing the non-convective scheme with a focus on the spectral properties of the iterative matrix and then employing a delicate perturbation analysis for the convective scheme. The work achieves the weakly divergence-free velocity (strongly divergence-free for divergence-free finite element spaces) and the rigorous stability and error analysis when the iterations converge The three-dimensional numerical tests confirm that this new method can effectively treat high Reynolds numbers with only a few iterations per time, where the convectional projection method and the iterative projection method with the explicit convection would fail.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 5","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145210928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A two-grid method with dispersion matching for finite-element Helmholtz problems","authors":"Christiaan C. Stolk","doi":"10.1007/s10444-025-10256-6","DOIUrl":"10.1007/s10444-025-10256-6","url":null,"abstract":"<div><p>This work is about a new two-level solver for Helmholtz equations discretized by finite elements. The method is inspired by two-grid methods for finite-difference Helmholtz problems as well as by previous work on two-level domain-decomposition methods. For the coarse-level discretization, a compact-stencil finite-difference method is used that minimizes dispersion errors. The smoother involves a domain-decomposition solver applied to a complex-shifted Helmholtz operator. Local Fourier analysis shows the method is convergent if the number of degrees of freedom per wavelength is larger than some lower bound that depends on the order, e.g., more than 8 for order 4. In numerical tests, with problem sizes up to 80 wavelenghts, convergence was fast, and almost independent of problem size unlike what is observed for conventional methods. Analysis and comparison with dispersion-error data shows that, for good convergence of a two-grid method for Helmholtz problems, it is essential that fine- and coarse-level dispersion relations closely match.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 5","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10256-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145073981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A primal-dual adaptive finite element method for total variation minimization","authors":"Martin Alkämper,&nbsp;Stephan Hilb,&nbsp;Andreas Langer","doi":"10.1007/s10444-025-10254-8","DOIUrl":"10.1007/s10444-025-10254-8","url":null,"abstract":"<div><p>Based on previous work, we extend a primal-dual semi-smooth Newton method for minimizing a general <span>(varvec{L^1})</span>-<span>(varvec{L^2})</span>-<span>(varvec{TV})</span> functional over the space of functions of bounded variations by adaptivity in a finite element setting. For automatically generating an adaptive grid, we introduce indicators based on a-posteriori error estimates. Further, we discuss data interpolation methods on unstructured grids in the context of image processing and present a pixel-based interpolation method. The efficiency of our derived adaptive finite element scheme is demonstrated on image inpainting and the task of computing the optical flow in image sequences. In particular, for optical flow estimation, we derive an adaptive finite element coarse-to-fine scheme which allows resolving large displacements and speeds up the computing time significantly.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 5","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10254-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144888091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two-level discretization of the 3D stationary Navier–Stokes equations with damping based on a difference finite element method","authors":"Qi Zhang,&nbsp;Pengzhan Huang","doi":"10.1007/s10444-025-10255-7","DOIUrl":"10.1007/s10444-025-10255-7","url":null,"abstract":"<div><p>A difference finite element method based on the mixed finite element pair <span>(((P_1^b,P_1^b,P_1) times (P_1,P_1,P_1)))</span>-<span>((P_1 times P_0))</span> is presented for the three-dimensional stationary Navier–Stokes equations with damping. Moreover, based on this proposed method, a two-level discretization is constructed, which involves solving a problem of the Navier–Stokes equations with damping on coarse mesh with mesh sizes <i>H</i> and <span>(mathcal {T})</span>, and a general Stokes problem on fine mesh with mesh sizes <span>(h = O(H^2))</span> and <span>(tau = O(mathcal {T}^2))</span>. This two-level difference finite element method provides an approximate solution with the same convergence rate as the difference finite element solution, which involves solving a problem of the Navier–Stokes equations with damping on fine mesh with mesh sizes <i>h</i> and <span>(tau )</span>. Hence, it can save a large amount of computational time. Finally, all computational results support the theoretical analysis and show the effectiveness of the two-level difference finite element method for solving the considered problem.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144832090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An adaptive finite element DtN method for the acoustic-elastic interaction problem in periodic structures","authors":"Lei Lin,&nbsp;Junliang Lv","doi":"10.1007/s10444-025-10253-9","DOIUrl":"10.1007/s10444-025-10253-9","url":null,"abstract":"<div><p>Consider a time-harmonic acoustic plane wave incident onto an elastic body with an unbounded periodic surface. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid air/fluid of constant mass density, while the elastic body is assumed to be isotropic and linear. By introducing the Dirichlet-to-Neumann (DtN) operators for acoustic and elastic waves simultaneously, the model is formulated as an acoustic-elastic interaction problem in periodic structures. Based on a duality argument, an a posteriori error estimate is derived for the associated truncated finite element approximation. The a posteriori error estimate consists of the finite element approximation error and the truncation error of two different DtN operators, where the latter decays exponentially with respect to the truncation parameter. Based on the a posteriori error, an adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem in periodic structures. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithm.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144832051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exponential decay and numerical treatment for mixture problem with Fourier law and frictional damping","authors":"Mauro L. Santos,&nbsp;Anderson J. A. Ramos,&nbsp;Anderson D. S. Campelo","doi":"10.1007/s10444-025-10250-y","DOIUrl":"10.1007/s10444-025-10250-y","url":null,"abstract":"<div><p>This study investigates a finite difference numerical scheme to analyze the impact of the effects caused by the strong coupling of Fourier’s law on the solutions of the equations of motion of a mixture of two one-dimensional linear isotropic elastic materials with frictional damping. We first prove the existence of solutions and exponential stability. In the sequence, we analyze the semi-discrete problem in finite differences and we use the energy method to prove the exponential stabilization of the corresponding semi-discrete system. The positivity of the numerical energy is also proved, and we present a fully discrete finite difference scheme that combines explicit and implicit integration methods. Finally, numerical simulations are given to confirm the theoretical results and show the efficiency of the proposed scheme.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uniform error bounds of a nested Picard iterative integrator for the Klein-Gordon-Zakharov system in the subsonic limit regime","authors":"Jiyong Li","doi":"10.1007/s10444-025-10251-x","DOIUrl":"10.1007/s10444-025-10251-x","url":null,"abstract":"<div><p>We propose a nested Picard iterative integrator Fourier pseudo-spectral (NPI-FP) method and establish the uniform error bounds for the Klein-Gordon-Zakharov system (KGZS) with <span>(varepsilon in (0, 1])</span> being a small parameter. In the subsonic limit regime (<span>(0 &lt; varepsilon ll 1)</span>), the solution of KGZS propagates waves with wavelength <span>(O(varepsilon ))</span> in time and amplitude at <span>(O(varepsilon ^{alpha ^dagger } ))</span> with <span>(alpha ^dagger =min {alpha ,beta +1,2})</span>, where <span>(alpha )</span> and <span>(beta )</span> describe the incompatibility between the initial data of the KGZS and the limiting equation as <span>(varepsilon rightarrow 0^+)</span> and satisfy <span>(alpha ge 0)</span>, <span>(beta +1ge 0)</span>. The oscillation in time becomes the main difficulty in constructing numerical schemes and making the corresponding error analysis for KGZS in this regime. In this paper, firstly, in order to overcome the difficulty of controlling nonlinear terms, we transform the KGZS into a system with higher derivative. Using the technique of nested Picard iteration, we construct a new time semi-discretization scheme and obtain the error estimates of semi-discretization with the bounds at <span>(O(min {tau ,tau ^2/varepsilon ^{1-alpha ^*}}))</span> for <span>(beta ge 0)</span> where <span>(alpha ^*=min {1,alpha ,1+beta })</span> and <span>(tau )</span> is time step. Hence, we get uniformly second-order error bounds at <span>(O(tau ^{2}))</span> when <span>(alpha ge 1)</span> and <span>(beta ge 0)</span>, and uniformly accurate first-order error estimates for any <span>(alpha ge 0)</span> and <span>(beta ge 0)</span>. We also give full discretization by Fourier pseudo-spectral method and obtain the error bounds at <span>(O(h^{sigma +2}+min {tau ,tau ^2/varepsilon ^{1-alpha ^*}}))</span>, where <i>h</i> is mesh size and <span>(sigma )</span> depends on the regularity of the solution. Hence, we get uniformly accurate spatial spectral order for any <span>(alpha ge 0)</span> and <span>(beta ge 0)</span>. Our numerical results support the error estimates. Surprisingly, our numerical results suggest a better error bound at <span>(O(h^{sigma +2}+ varepsilon ^qtau ^{2}))</span> for a certain <span>(qin mathbb {R})</span>.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Galerkin neural network-POD for acoustic and electromagnetic wave propagation in parametric domains","authors":"Philipp Weder,&nbsp;Mariella Kast,&nbsp;Fernando Henríquez,&nbsp;Jan S. Hesthaven","doi":"10.1007/s10444-025-10245-9","DOIUrl":"10.1007/s10444-025-10245-9","url":null,"abstract":"<div><p>We investigate reduced order models for acoustic and electromagnetic wave problems in parametrically defined domains. The parameter-to-solution maps are approximated following the so-called Galerkin POD-NN method, which combines the construction of a reduced basis via proper orthogonal decomposition (POD) with neural networks (NNs). As opposed to the standard reduced basis method, this approach allows for the swift and efficient evaluation of reduced order solutions for any given parametric input. As is customary in the analysis of problems in random or parametrically defined domains, we start by transporting the formulation to a reference domain. This yields a parameter-dependent variational problem set on parameter-independent functional spaces. In particular, we consider affine-parametric domain transformations characterized by a high-dimensional, possibly countably infinite, parametric input. To keep the number of evaluations of the high-fidelity solutions manageable, we propose using low-discrepancy sequences to sample the parameter space efficiently. Then, we train an NN to learn the coefficients in the reduced representation. This approach completely decouples the offline and online stages of the reduced basis paradigm. Numerical results for the three-dimensional Helmholtz and Maxwell equations confirm the method’s accuracy up to a certain barrier and show significant gains in online speed-up compared to the traditional Galerkin POD method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10245-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Energy-stable and efficient finite element schemes for the Shliomis model of ferrofluid flows","authors":"Guo-Dong Zhang,&nbsp;Kejia Pan,&nbsp;Xiaoming He,&nbsp;Xiaofeng Yang","doi":"10.1007/s10444-025-10249-5","DOIUrl":"10.1007/s10444-025-10249-5","url":null,"abstract":"<div><p>In this paper, we aim to design two energy-stable and efficient finite element schemes for simulating the ferrofluid flows based on the well-known Shliomis model. The model is a highly nonlinear, coupled, multi-physics system, consisting of the Navier–Stokes equations, magnetostatic equation, and magnetization field equation. We propose two reliable numerical algorithms with the following desired features: linearity and unconditional energy stability. Several key techniques are used to achieve the required features, including the auxiliary variable method, consistent terms method, prediction-correction method, and semi-implicit stabilization method. The first scheme is based on a hybrid continuous/discontinuous finite elements spatial approximation, and the second utilizes decoupled continuous finite element spatial discretization. We have rigorously demonstrated that the proposed schemes are unconditionally energy stable and carried out extensive numerical simulations to illustrate the accuracy and stability of the developed schemes, as well as some interesting controllable characteristics of the ferrofluid flows.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}