Applied Numerical Mathematics最新文献

{"title":"A high-order accurate unconditionally stable bound-preserving numerical scheme for the Cahn-Hilliard-Navier-Stokes equations","authors":"Yali Gao ,&nbsp;Daozhi Han ,&nbsp;Sayantan Sarkar","doi":"10.1016/j.apnum.2025.06.004","DOIUrl":"10.1016/j.apnum.2025.06.004","url":null,"abstract":"<div><div>A high order numerical method is developed for solving the Cahn-Hilliard-Navier-Stokes equations with the Flory-Huggins potential. The scheme is based on the <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> finite element with mass lumping on rectangular grids, the second-order convex splitting method and the pressure correction method. The unique solvability, unconditional stability, and bound-preserving properties are rigorously established. The key for bound-preservation is the discrete <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> estimate of the singular potential. Ample numerical experiments are performed to validate the desired properties of the proposed numerical scheme.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 96-111"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263514","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 efficient multi projection methods for systems of Fredholm integral equations with mixed weakly singular kernels: A superconvergence approach","authors":"Krishna Murari Malav ,&nbsp;Kapil Kant ,&nbsp;Joydip Dhar ,&nbsp;Samiran Chakraborty","doi":"10.1016/j.apnum.2025.06.006","DOIUrl":"10.1016/j.apnum.2025.06.006","url":null,"abstract":"<div><div>In this article, we develop the multi-Galerkin and iterated multi-Galerkin methods to solve systems of second-kind linear Fredholm integral equations (FIEs) with smooth and mixed weakly singular kernels. First, we develop the mathematical formulation of the multi-Galerkin and iterated multi-Galerkin methods using piecewise polynomial approximations to solve such systems and obtain superconvergence results. These methods transform the linear system of FIEs into corresponding matrix equations. We derive error estimates and obtain the convergence analysis. We prove that the convergence rates for the multi-Galerkin method are <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>3</mn><mi>r</mi></mrow></msup><mo>)</mo></math></span> for smooth kernels and <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>r</mi><mo>−</mo><mi>α</mi></mrow></msup><mi>log</mi><mo>⁡</mo><mi>h</mi><mo>)</mo></math></span> for mixed weakly singular kernels, where <em>r</em> denote the degree of the piecewise polynomials, <em>h</em> is the norm of partitions and <span><math><mi>α</mi><mo>=</mo><munder><mi>max</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mo>⁡</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span>. Moreover, we establish that the iterated multi-Galerkin method achieves improved convergence rates of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>4</mn><mi>r</mi></mrow></msup><mo>)</mo></math></span> for smooth kernels and <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>2</mn><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>h</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> for mixed weakly singular kernels. Hence, the results show that the iterated multi-Galerkin method improves the multi-Galerkin method. Finally, the theoretical results are validated through the numerical examples.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 172-189"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322646","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":"Analysis of a divergence-free element-free Galerkin method for the Navier-Stokes equations","authors":"Xiaolin Li ,&nbsp;Haiyun Dong","doi":"10.1016/j.apnum.2025.06.002","DOIUrl":"10.1016/j.apnum.2025.06.002","url":null,"abstract":"<div><div>In this paper, an efficient divergence-free element-free Galerkin (DFEFG) method is proposed for the numerical analysis of the incompressible Navier-Stokes equations. In this method, a divergence-free moving least squares (DFMLS) approximation is used to obtain the meshless approximation of the divergence-free velocity field. The properties, stability and error of the DFMLS approximation are analyzed firstly, and then the stability and error estimation of the DFEFG method are derived theoretically. Finally, numerical results demonstrate the efficiency of the proposed methods and verify the theoretical analysis.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 73-95"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263513","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 surface mesh DG-VEM for elliptic membrane shell model","authors":"Qian Yang ,&nbsp;Xiaoqin Shen ,&nbsp;Jikun Zhao ,&nbsp;Zhiming Gao","doi":"10.1016/j.apnum.2025.06.014","DOIUrl":"10.1016/j.apnum.2025.06.014","url":null,"abstract":"<div><div>Elliptic membrane shell (EMS), characterized by a system with complex variable coefficients on a surface, poses significant challenges for numerical discretization. In this paper, leveraging the differing regularity of displacement components, we propose a discontinuous Galerkin virtual element method (DG-VEM) for the EMS model. Specifically, we construct <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span>-continuous virtual element spaces for the first two components, whereas the third component is discretized on each element using a polynomial of degree <em>l</em>, with no continuity enforced across element boundaries. This method offers high mesh flexibility, eliminates the need for explicit basis function expressions, and improves accuracy to achieve convergence of any desired order. Furthermore, we establish the existence, uniqueness, stability, and convergence of the numerical solution, along with rigorous error estimates. Several numerical examples are presented to test the convergence and stability of the DG-VEM. Additionally, we demonstrate the method's adaptability to diverse grid subdivisions and show that, for comparable error levels, the DG-VEM for the EMS model requires significantly fewer degrees of freedom than traditional finite element methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 298-318"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517713","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 new error analysis of an explicit skeletal discontinuous Galerkin scheme for time-dependent Maxwell equations","authors":"Achyuta Ranjan Dutta Mohapatra,&nbsp;Bhupen Deka","doi":"10.1016/j.apnum.2025.05.011","DOIUrl":"10.1016/j.apnum.2025.05.011","url":null,"abstract":"<div><div>This article focuses on the convergence analysis of second-order time-dependent Maxwell equations in conducting and non-conducting media. Firstly, the spatial discretization is made using a skeletal discontinuous Galerkin method, and then some essential tools are established to deduce the stability and error estimates. Under suitable regularity assumptions on initial data, stability and optimal convergence of the error are proved for the semi-discrete problem in a discretely defined <span><math><mi>H</mi><mtext>(curl)</mtext></math></span>-norm. Next, temporal discretization is applied to the semi-discrete system by using the explicit Leap-frog schemes and optimal error bounds <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>, <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> is the degree of polynomial approximation, are achieved in the discrete energy norm. Finally, computational experiments are performed to validate the theoretical conclusions. Error analysis of explicit fully discrete schemes in polygonal/polyhedral meshes for the second-order Maxwell equations in a conducting media is missing in the literature, and we intend to fill this gap in the current article.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 18-42"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144221744","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":"Localization of tumor through a non-conventional numerical shape optimization technique","authors":"Julius Fergy Tiongson Rabago","doi":"10.1016/j.apnum.2025.06.005","DOIUrl":"10.1016/j.apnum.2025.06.005","url":null,"abstract":"<div><div>This paper presents a method for estimating the shape and location of an embedded tumor using shape optimization techniques, specifically through the coupled complex boundary method. The inverse problem—characterized by a measured temperature profile and corresponding heat flux (e.g., from infrared thermography)—is reformulated as a complex boundary value problem with a complex Robin boundary condition, thereby simplifying its over-specified nature. The geometry of the tumor is identified by optimizing an objective functional that depends on the imaginary part of the solution throughout the domain. The shape derivative of the functional is derived through shape sensitivity analysis. An iterative algorithm is developed to numerically recover the tumor shape, based on the Riesz representative of the gradient and implemented using the finite element method. In addition, the mesh sensitivity of the finite element solution to the state problem is analyzed, and bounds are established for its variation with respect to mesh deformation and its gradient. Numerical examples are presented to validate the theoretical results and to demonstrate the accuracy and effectiveness of the proposed method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 135-171"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307834","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-level convergence order accelerators of iterative methods for nonlinear problems","authors":"Alicia Cordero ,&nbsp;Renso V. Rojas-Hiciano ,&nbsp;Juan R. Torregrosa ,&nbsp;Maria P. Vassileva","doi":"10.1016/j.apnum.2025.07.003","DOIUrl":"10.1016/j.apnum.2025.07.003","url":null,"abstract":"<div><div>We present an efficient strategy to increase, under certain conditions, the order of convergence of iterative methods to solve nonlinear systems of equations. We analytically compare the new accelerator with others and establish the conditions under which this technique is more efficient. We perform an analysis of the efficiency of some one-step accelerators that increase the convergence order by two units. New concepts about efficiency are introduced which allow us to compare different iterative schemes from other points of view. We demonstrate that our proposal is a good alternative to the existing ones. As a consequence, we propose two new maximally efficient, damped Newton-Traub type schemes of order 5 and 6. These are an improvement of two other maximally efficient methods. Their numerical performance is better than that of known methods of the same order, and we find that it is a very economical way to achieve high order. Some numerical examples confirm the theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 390-411"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580373","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":"Exponentially accurate spectral Monte Carlo method for linear PDEs and their error estimates","authors":"Jiaying Feng,&nbsp;Changtao Sheng,&nbsp;Chenglong Xu","doi":"10.1016/j.apnum.2025.06.010","DOIUrl":"10.1016/j.apnum.2025.06.010","url":null,"abstract":"<div><div>This paper introduces a spectral Monte Carlo iterative method (SMC) for solving linear Poisson and parabolic equations driven by <em>α</em>-stable Lévy processes with <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, which was initially proposed and developed by Gobet and Maire in their pioneering works ((2004) <span><span>[24]</span></span>, and (2005) <span><span>[25]</span></span>) for the case <span><math><mi>α</mi><mo>=</mo><mn>2</mn></math></span>. The novel method effectively integrates multiple computational techniques, including the interpolation based on generalized Jacobi functions (GJFs), space-time spectral methods, control variates techniques, and a novel walk-on-spheres method (WOS). The exponential convergence of the error bounds is rigorously established through finite iterations for both Poisson and parabolic equations involving the integral fractional Laplacian operator. Remarkably, the proposed space-time spectral Monte Carlo method (ST-SMC) for the parabolic equation is unified for both <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> and <span><math><mi>α</mi><mo>=</mo><mn>2</mn></math></span>. Extensive numerical results are provided to demonstrate the spectral accuracy and efficiency of the proposed method, thereby validating the theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 278-297"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489532","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":"Generalization error analysis of deep backward dynamic programming for solving nonlinear PDEs","authors":"Du Ouyang,&nbsp;Jichang Xiao,&nbsp;Xiaoqun Wang","doi":"10.1016/j.apnum.2025.06.013","DOIUrl":"10.1016/j.apnum.2025.06.013","url":null,"abstract":"<div><div>We explore the application of the quasi-Monte Carlo (QMC) method in deep backward dynamic programming (DBDP) <span><span>[1]</span></span> for numerically solving high-dimensional nonlinear partial differential equations (PDEs). Our study focuses on examining the generalization error as a component of the total error in the DBDP framework, discovering that the rate of convergence for the generalization error is influenced by the choice of sampling methods. Specifically, for a given batch size <em>m</em>, the generalization error under QMC methods exhibits a convergence rate of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span>, where <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span>. Our theoretical analysis shows that the generalization error under QMC methods achieves a higher order of convergence than their MC counterparts. Numerical experiments demonstrate that QMC indeed surpasses MC in delivering solutions that are both more precise and stable.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 319-339"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144522669","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":"Note on an inertial projection-based approach for solving extended general quasi-variational inequalities and its convergence analysis","authors":"Saudia Jabeen ,&nbsp;Siegfried Macías ,&nbsp;Saleem Ullah ,&nbsp;Muhammad Aslam Noor ,&nbsp;Jorge E. Macías-Díaz","doi":"10.1016/j.apnum.2025.06.012","DOIUrl":"10.1016/j.apnum.2025.06.012","url":null,"abstract":"<div><div>In this work, an inertial projection-based method is proposed to find approximate solutions to a new class of quasi-variational inequalities. The approach utilizes projection techniques to develop an iterative scheme, and its convergence properties are rigorously examined under appropriate conditions. Various specific cases are derived from the general framework, highlighting the adaptability of the proposed method. The comparative performance of this approach with existing techniques remains an open area for exploration. It is expected that the theoretical framework and techniques proposed in this study will stimulate further research in this domain.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 190-198"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338385","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}