Computers & Mathematics with Applications最新文献

{"title":"Vectorized implementation of primal hybrid FEM in MATLAB","authors":"Harish Nagula Mallesham ,&nbsp;Kamana Porwal ,&nbsp;Jan Valdman ,&nbsp;Sanjib Kumar Acharya","doi":"10.1016/j.camwa.2024.12.017","DOIUrl":"10.1016/j.camwa.2024.12.017","url":null,"abstract":"<div><div>We present efficient MATLAB implementations of the lowest-order primal hybrid finite element method (FEM) for linear second-order elliptic and parabolic problems with mixed boundary conditions in two spatial dimensions. We employ backward Euler and the Crank-Nicolson finite difference scheme for the complete discrete setup of the parabolic problem. All the codes presented are fully vectorized using matrix-wise array operations. Numerical experiments are conducted to show the performance of the software.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 144-165"},"PeriodicalIF":2.9,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142911856","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":"Semi-analytical algorithm for quasicrystal patterns","authors":"Keyue Sun,&nbsp;Xiangjie Kong,&nbsp;Junxiang Yang","doi":"10.1016/j.camwa.2024.12.016","DOIUrl":"10.1016/j.camwa.2024.12.016","url":null,"abstract":"<div><div>To efficiently simulate the quasicrystal patterns, we present a multi-stage semi-analytically algorithm. Utilizing the operator splitting strategy, we first split the original equation into three subproblems. A second-order five-stage scheme consists of solving four nonlinear ordinary differential equations with half time step and solving a linear partial differential equation with full time step. Using the methods of separation of variables, the nonlinear ODEs have analytical solutions. The linear PDE can also be analytically solved by using the Fourier-spectral method in space. In this sense, our proposed is semi-analytical because we only adopt an approximation in time. In each time step, we only need to compute several analytically solutions in a step-by-step manner. Therefore, the algorithm will be highly efficient and the simulation can be easily implemented. The performance and high efficiency of our proposed algorithm are verified via several simulations. To facilitate the interested readers to develop related researches, a MATLAB code for generating 12-fold quasicrystal patterns is provided in Appendix. We also share the computational code on Code Ocean platform, please refer to <span><span>https://doi.org/10.24433/CO.6028082.v1</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 130-143"},"PeriodicalIF":2.9,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142911894","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":"On full linear convergence and optimal complexity of adaptive FEM with inexact solver","authors":"Philipp Bringmann,&nbsp;Michael Feischl,&nbsp;Ani Miraçi,&nbsp;Dirk Praetorius,&nbsp;Julian Streitberger","doi":"10.1016/j.camwa.2024.12.013","DOIUrl":"10.1016/j.camwa.2024.12.013","url":null,"abstract":"<div><div>The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computation time. To this end, algorithmically, the standard adaptive finite element method (AFEM) integrates an inexact solver and nested iterations with discerning stopping criteria balancing the different error components. The analysis ensuring optimal convergence order of AFEM with respect to the overall computational cost critically hinges on the concept of R-linear convergence of a suitable quasi-error quantity. This work tackles several shortcomings of previous approaches by introducing a new proof strategy. Previously, the analysis of the algorithm required several parameters to be fine-tuned. This work leaves the classical reasoning and introduces a summability criterion for R-linear convergence to remove restrictions on those parameters. Second, the usual assumption of a (quasi-)Pythagorean identity is replaced by the generalized notion of quasi-orthogonality from Feischl (2022) <span><span>[22]</span></span>. Importantly, this paves the way towards extending the analysis of AFEM with inexact solver to general inf-sup stable problems beyond the energy minimization setting. Numerical experiments investigate the choice of the adaptivity parameters.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 102-129"},"PeriodicalIF":2.9,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142911902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unconditional superconvergence analysis of a novel energy dissipation nonconforming Crank-Nicolson FEM for Sobolev equations with high order Burgers' type nonlinearity","authors":"Tiantian Liang ,&nbsp;Dongyang Shi","doi":"10.1016/j.camwa.2024.12.010","DOIUrl":"10.1016/j.camwa.2024.12.010","url":null,"abstract":"<div><div>A novel energy dissipation Crank-Nicolson (C-N) fully discrete scheme is established by low order nonconforming <span><math><mi>E</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi><mi>o</mi><mi>t</mi></mrow></msubsup></math></span> element for solving the Sobolev equations with high order Burgers' type nonlinearity. Firstly, the boundedness of the discrete solution in the broken <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm is achieved directly by the energy dissipation property without using the known time-space splitting technique in the existing literatures, and its well-posedness is demonstrated by the Brouwer fixed point theorem. Secondly, by utilizing the special characters of nonconforming <span><math><mi>E</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi><mi>o</mi><mi>t</mi></mrow></msubsup></math></span> element, the unconditional superclose result of order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> in the broken <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm is gained strictly with no restrictions between the spatial partition parameter <em>h</em> and the time step <em>τ</em>. Moreover, the corresponding global superconvergent error estimate of order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> is proved by applying an interpolation post-processing approach. Thirdly, an application to some different finite elements and nonlinear PDEs is discussed, which shows that the proposed scheme and the analysis presented herein can be considered as a general framework to cope with. Lastly, the theoretical results are validated by four numerical examples.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 86-101"},"PeriodicalIF":2.9,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142884407","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":"Energy-preserving RERK-FEM for the regularized logarithmic Schrödinger equation","authors":"Changhui Yao ,&nbsp;Lei Li ,&nbsp;Huijun Fan ,&nbsp;Yanmin Zhao","doi":"10.1016/j.camwa.2024.12.009","DOIUrl":"10.1016/j.camwa.2024.12.009","url":null,"abstract":"<div><div>A high-order implicit–explicit (IMEX) finite element method with energy conservation is constructed to solve the regularized logarithmic Schrödinger equation (RLogSE) with a periodic boundary condition. The discrete scheme consists of the relaxation-extrapolated Runge–Kutta (RERK) method in the temporal direction and the finite element method in the spatial direction. Choosing a proper relaxation parameter for the RERK method is the key technique for energy conservation. The optimal error estimates in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm are provided without any restrictions between time step size <em>τ</em> and mesh size <em>h</em> by temporal–spatial splitting technology. Numerical examples are given to demonstrate the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 76-85"},"PeriodicalIF":2.9,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142884406","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 study of an efficient numerical method for solving the generalized fractional reaction-diffusion model involving a distributed-order operator along with stability analysis","authors":"Muhammad Suliman ,&nbsp;Muhammad Ibrahim ,&nbsp;Ebrahem A. Algehyne ,&nbsp;Vakkar Ali","doi":"10.1016/j.camwa.2024.12.006","DOIUrl":"10.1016/j.camwa.2024.12.006","url":null,"abstract":"<div><div>In this manuscript, we study a generalized fractional reaction-diffusion model involving a distributed-order operator. An efficient hybrid approach is proposed to solve the presented model. The <em>L</em>1 approximation is utilized to discretize the time variable, while the mixed finite element method is employed for spatial discretization. A detailed error and stability analysis of the proposed method is provided. Furthermore, we prove that the computational accuracy achieved by the proposed approach is of order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>Δ</mi><mi>t</mi><mo>)</mo></mrow><mrow><mn>3</mn><mo>−</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>max</mi></mrow></msub></mrow></msup><mo>)</mo></math></span>. To validate and evaluate the numerical approach, three numerical experiments are conducted, with results presented through graphs and tables.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 61-75"},"PeriodicalIF":2.9,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142884409","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":"Proving the stability estimates of variational least-squares kernel-based methods","authors":"Meng Chen ,&nbsp;Leevan Ling ,&nbsp;Dongfang Yun","doi":"10.1016/j.camwa.2024.12.008","DOIUrl":"10.1016/j.camwa.2024.12.008","url":null,"abstract":"<div><div>Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 46-60"},"PeriodicalIF":2.9,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong approximation of the time-fractional Cahn–Hilliard equation driven by a fractionally integrated additive noise","authors":"Mariam Al-Maskari,&nbsp;Samir Karaa","doi":"10.1016/j.camwa.2024.12.007","DOIUrl":"10.1016/j.camwa.2024.12.007","url":null,"abstract":"<div><div>In this article, we present a numerical scheme for solving a time-fractional stochastic Cahn–Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and a fractional time-integral noise of order <span><math><mi>γ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. Our numerical approach combines a piecewise linear finite element method in space with a convolution quadrature in time, designed to handle both time-fractional operators, along with the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-projection for the noise. We conduct a detailed analysis of both spatially semidiscrete and fully discrete schemes, deriving strong convergence rates through energy-based arguments. The solution's temporal Hölder continuity played a key role in the error analysis. Unlike the stochastic Allen–Cahn equation, the inclusion of the unbounded elliptic operator in front of the cubic nonlinearity in our model added complexity and challenges to the error analysis. To address these challenges, we introduce novel techniques and refined error estimates. We conclude with numerical examples that validate our theoretical findings.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 28-45"},"PeriodicalIF":2.9,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841376","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 high-precision numerical method based on spectral deferred correction for solving the time-fractional Allen-Cahn equation","authors":"Jing Wang,&nbsp;Xuejuan Chen,&nbsp;Jinghua Chen","doi":"10.1016/j.camwa.2024.11.034","DOIUrl":"10.1016/j.camwa.2024.11.034","url":null,"abstract":"<div><div>This paper presents a high-precision numerical method based on spectral deferred correction (SDC) for solving the time-fractional Allen-Cahn equation. In the temporal direction, we establish a stabilized variable-step <em>L</em>1 semi-implicit scheme which satisfies the discrete variational energy dissipation law and the maximum principle. Through theoretical analysis, we prove that this numerical scheme is convergent and unconditionally stable. In the spatial direction, we apply the Fourier-Galerkin spectral method for discretization and conduct an error analysis of the fully discretized scheme. Since the stabilized variable-step <em>L</em>1 semi-implicit scheme is only of first-order accuracy in the time direction, to improve the accuracy, we combine explicit and implicit schemes (linear terms are handled implicitly, while nonlinear terms are handled explicitly) to establish a stabilized semi-implicit spectral deferred correction scheme. Finally, we verify the validity and feasibility of the numerical scheme through numerical examples.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 1-27"},"PeriodicalIF":2.9,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841379","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":"Meshfree methods for nonlinear equilibrium radiation diffusion equation with interface and discontinuous coefficient","authors":"Haowei Liu,&nbsp;Zhiyong Liu,&nbsp;Qiuyan Xu,&nbsp;Jiye Yang","doi":"10.1016/j.camwa.2024.11.029","DOIUrl":"10.1016/j.camwa.2024.11.029","url":null,"abstract":"<div><div>The partial differential equation describing equilibrium radiation diffusion is strongly nonlinear, which has been widely utilized in various fields such as astrophysics and others. The equilibrium radiation diffusion equation usually appears over multiple complicated domains, and the material characteristics vary between each domain. The diffusion coefficient near the interface is discontinuous. In this paper, the equilibrium radiation diffusion equation with discontinuous diffusion coefficient will be solved numerically by the unsymmetric radial basis function collocation method. The energy term <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> is linearized by utilizing the Picard-Newton and Richtmyer linearization methods on the basis of the fully implicit scheme discretization. And the successive permutation iteration and direct linearization methods are applied to linearize the diffusion terms. The accuracy of the proposed methods is proved by numerical experiments for regular and irregular domains with different types of interfaces.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"178 ","pages":"Pages 108-135"},"PeriodicalIF":2.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142758959","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}