Pub Date : 2026-10-01Epub Date: 2026-02-26DOI: 10.1016/j.cam.2026.117483
Xuyang Na
In this paper, we develop a two-dimensional Robin-type nonoverlapping domain decomposition (DD) method for the nonconforming Crouzeix-Raviart (CR) finite element. In a previous study, Qin and Xu introduced a Robin-type DD iterative method for the CR finite element and proved its convergence rate is , where h is the mesh size, H is the diameter of subdomains and N is the so-called winding number. In this paper, we improve the result and design a quasi-optimal preconditioned method. It is proved that the condition number of the preconditioned system grows only as . Especially, for a special case of discontinuous coefficients arranged in a checkerboard pattern, our method converges fast. Numerical experiments are performed to verify our conclusions.
{"title":"A Robin-type domain decomposition method for Crouzeix-Raviart finite element discretizations","authors":"Xuyang Na","doi":"10.1016/j.cam.2026.117483","DOIUrl":"10.1016/j.cam.2026.117483","url":null,"abstract":"<div><div>In this paper, we develop a two-dimensional Robin-type nonoverlapping domain decomposition (DD) method for the nonconforming Crouzeix-Raviart (CR) finite element. In a previous study, Qin and Xu introduced a Robin-type DD iterative method for the CR finite element and proved its convergence rate is <span><math><mrow><mn>1</mn><mo>−</mo><msup><mi>C</mi><mi>N</mi></msup><msup><mi>h</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>H</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>, where <em>h</em> is the mesh size, <em>H</em> is the diameter of subdomains and <em>N</em> is the so-called winding number. In this paper, we improve the result and design a quasi-optimal preconditioned method. It is proved that the condition number of the preconditioned system grows only as <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>log</mi><mfrac><mi>H</mi><mi>h</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup></math></span>. Especially, for a special case of discontinuous coefficients arranged in a checkerboard pattern, our method converges fast. Numerical experiments are performed to verify our conclusions.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117483"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386635","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}
Pub Date : 2026-10-01Epub Date: 2026-02-28DOI: 10.1016/j.cam.2026.117538
B. El-Sobky, G. Ashry
In this paper, we introduce an algorithm to solve a nonlinear finite minimax problem. This algorithm is based on converting minimax problems to differentiable optimization problems with constraints. This allows using a slack variable with a penalty method to convert the differentiable optimization problems with constraints to an unconstrained optimization problem with bound-on variables. To solve the unconstrained optimization problem with bound-on variables, Newton’s interior-point method is used. But Newton’s method is local and so it may not converge if the starting point is far away from a stationary point. To guarantee convergence from any starting point to the stationary point, a trust-region globalization strategy is used.
A global convergence theory for the proposed algorithm is introduced under three standard assumptions.
Finally, numerical experiments are reported to indicate that the proposed algorithm performs efficiently in practice
{"title":"An interior-point trust-region algorithm to solve a finite nonlinear minimax problem","authors":"B. El-Sobky, G. Ashry","doi":"10.1016/j.cam.2026.117538","DOIUrl":"10.1016/j.cam.2026.117538","url":null,"abstract":"<div><div>In this paper, we introduce an algorithm to solve a nonlinear finite minimax problem. This algorithm is based on converting minimax problems to differentiable optimization problems with constraints. This allows using a slack variable with a penalty method to convert the differentiable optimization problems with constraints to an unconstrained optimization problem with bound-on variables. To solve the unconstrained optimization problem with bound-on variables, Newton’s interior-point method is used. But Newton’s method is local and so it may not converge if the starting point is far away from a stationary point. To guarantee convergence from any starting point to the stationary point, a trust-region globalization strategy is used.</div><div>A global convergence theory for the proposed algorithm is introduced under three standard assumptions.</div><div>Finally, numerical experiments are reported to indicate that the proposed algorithm performs efficiently in practice</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117538"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386904","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}
Pub Date : 2026-09-01Epub Date: 2026-01-22DOI: 10.1016/j.cam.2026.117377
Ruyun Chen, Hong Du
This work develops two matrix-based quadrature rules to compute the integrals containing products of two Bessel functions. By reformulating these integrals into a matrix framework and employing low-order derivatives of Bessel functions of the first kind in combination with integration by parts, we construct both a matrix-based asymptotic rule and a matrix-based Filon-type rule. Numerical experiments confirm the theoretical analysis and highlight the efficiency of the proposed rules.
{"title":"Efficient matrix-based quadrature rules for oscillatory integrals with products of two Bessel functions","authors":"Ruyun Chen, Hong Du","doi":"10.1016/j.cam.2026.117377","DOIUrl":"10.1016/j.cam.2026.117377","url":null,"abstract":"<div><div>This work develops two matrix-based quadrature rules to compute the integrals containing products of two Bessel functions. By reformulating these integrals into a matrix framework and employing low-order derivatives of Bessel functions of the first kind in combination with integration by parts, we construct both a matrix-based asymptotic rule and a matrix-based Filon-type rule. Numerical experiments confirm the theoretical analysis and highlight the efficiency of the proposed rules.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117377"},"PeriodicalIF":2.6,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079935","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}
Pub Date : 2026-09-01Epub Date: 2026-01-19DOI: 10.1016/j.cam.2026.117370
E. Aourir , H. Laeli Dastjerdi , M. Oudani
This work presents a new algorithm for solving a kind of Volterra delay integral equations of the third kind (VDIEs). Using the Tau method and generalized polynomial bases, our developed method is a robust approach for solving these equations. Specifically, we employ simple matrix operations to enhance the Tau approach. The underlying strategy leverages orthogonal polynomial bases to change the original equation into a matrix-vector form. Such a transformation makes the third-kind VDIEs easier to handle by turning them into a set of algebraic equations. Importantly, this method exhibits good stability, reduces memory usage, and is computationally cost-effective. The paper details the algorithm’s formulation and shows its capability to provide approximate polynomial solutions. We perform a thorough error estimation to check the method’s accuracy. To demonstrate its practical effectiveness, we use several numerical examples. The obtained results highlight the performance of the method and prove its alignment with theoretical error predictions. Furthermore, a comparative analysis with analytical solutions and alternative methods reaffirms the efficiency of the developed approach.
{"title":"An efficient Tau approach for solving a class of third-kind Volterra integral equations with proportional delays","authors":"E. Aourir , H. Laeli Dastjerdi , M. Oudani","doi":"10.1016/j.cam.2026.117370","DOIUrl":"10.1016/j.cam.2026.117370","url":null,"abstract":"<div><div>This work presents a new algorithm for solving a kind of Volterra delay integral equations of the third kind (VDIEs). Using the Tau method and generalized polynomial bases, our developed method is a robust approach for solving these equations. Specifically, we employ simple matrix operations to enhance the Tau approach. The underlying strategy leverages orthogonal polynomial bases to change the original equation into a matrix-vector form. Such a transformation makes the third-kind VDIEs easier to handle by turning them into a set of algebraic equations. Importantly, this method exhibits good stability, reduces memory usage, and is computationally cost-effective. The paper details the algorithm’s formulation and shows its capability to provide approximate polynomial solutions. We perform a thorough error estimation to check the method’s accuracy. To demonstrate its practical effectiveness, we use several numerical examples. The obtained results highlight the performance of the method and prove its alignment with theoretical error predictions. Furthermore, a comparative analysis with analytical solutions and alternative methods reaffirms the efficiency of the developed approach.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117370"},"PeriodicalIF":2.6,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039554","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}
Pub Date : 2026-09-01Epub Date: 2026-01-20DOI: 10.1016/j.cam.2026.117373
Wenfei Cao , Xicui Peng , Yang Chen , Jiahui Ji
In this paper, we study the tensor recovery problem from linear measurements corrupted by the ℓ1-bounded noise plus the adversarial noise with sparsity ratio ω. To handle this problem, we propose a novel least-absolute-deviation (LAD) loss minimization model based on low tubal-rank tensor decomposition. For the requirement of theoretical studies, we extend the mixed ℓ1/ℓ2-RIP and the ω-robustness to the tensor case, i.e., ℓ1/ℓ2-t-RIP and ω-t-robustness. Then leveraging these tools, we establish a reliable recovery guarantee for the proposed model, showing that when the sampling complexity reaches , the model’s optimal solution can robustly recover the original low tubal-rank tensor for any ω < 0.239. Moreover, we develop a subgradient descent algorithm to solve the proposed model and prove that it achieves geometrical convergence under appropriate initialization conditions. Finally, extensive experiments on the synthetic tensors and real video datasets are conducted to validate the exactness of the established theories and demonstrate the effectiveness of the proposed approach.
{"title":"Low tubal-rank tensor recovery with adversarial sparse noises","authors":"Wenfei Cao , Xicui Peng , Yang Chen , Jiahui Ji","doi":"10.1016/j.cam.2026.117373","DOIUrl":"10.1016/j.cam.2026.117373","url":null,"abstract":"<div><div>In this paper, we study the tensor recovery problem from linear measurements corrupted by the ℓ<sub>1</sub>-bounded noise plus the adversarial noise with sparsity ratio <em>ω</em>. To handle this problem, we propose a novel least-absolute-deviation (LAD) loss minimization model based on low tubal-rank tensor decomposition. For the requirement of theoretical studies, we extend the mixed ℓ<sub>1</sub>/ℓ<sub>2</sub>-RIP and the <em>ω</em>-<em>robustness</em> to the tensor case, i.e., ℓ<sub>1</sub>/ℓ<sub>2</sub>-<em>t</em>-RIP and <em>ω</em>-<em>t</em>-<em>robustness</em>. Then leveraging these tools, we establish a reliable recovery guarantee for the proposed model, showing that when the sampling complexity reaches <span><math><mrow><mi>O</mi><mo>(</mo><mrow><mo>(</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo>+</mo><mn>1</mn><mo>)</mo></mrow><msub><mi>n</mi><mn>3</mn></msub><mi>r</mi><mo>)</mo></mrow></math></span>, the model’s optimal solution can robustly recover the original low tubal-rank tensor for any <em>ω</em> < 0.239. Moreover, we develop a subgradient descent algorithm to solve the proposed model and prove that it achieves geometrical convergence under appropriate initialization conditions. Finally, extensive experiments on the synthetic tensors and real video datasets are conducted to validate the exactness of the established theories and demonstrate the effectiveness of the proposed approach.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117373"},"PeriodicalIF":2.6,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079951","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 develop efficient preconditioning techniques for distributed optimal control problems governed by partial differential equations with Caputo fractional derivative in time. By employing a discretize-then-optimize approach combining mixed all-at-once schemes of finite-difference for temporal and finite-element for spatial discretizations, we derive a large-scale and ill-conditioned Kronecker structured block two-by-two linear system with distinct pivot blocks. A block approximate factorization preconditioning method that is well-suited for approximating the Schur complement is considered by utilizing the so called matching strategy. A distinctive feature of the proposed preconditioner is its computational efficiency arising from its practical Schur complement-free implementation manner. Furthermore, the eigenvalues of the preconditioned system are demonstrated to lie within parameter-free positive real intervals, ensuring fast convergence independent of problem parameters under Krylov subspace acceleration. Motivated by the inherent block-Toeplitz structures, circulant-based inexact variants of the proposed preconditioner are explored and implemented within diagonalization strategies by fast Fourier transformation (FFT). Numerical experiments are conducted to validate the effectiveness and robustness of our proposed preconditioners compared with some optimal preconditioning strategies.
{"title":"Efficient preconditioning techniques for time-fractional PDE-constrained optimization problems","authors":"Zhao-Zheng Liang , Guo-Feng Zhang , Lei Zhang , Mu-Zheng Zhu","doi":"10.1016/j.cam.2026.117348","DOIUrl":"10.1016/j.cam.2026.117348","url":null,"abstract":"<div><div>In this paper, we develop efficient preconditioning techniques for distributed optimal control problems governed by partial differential equations with Caputo fractional derivative in time. By employing a discretize-then-optimize approach combining mixed all-at-once schemes of finite-difference for temporal and finite-element for spatial discretizations, we derive a large-scale and ill-conditioned Kronecker structured block two-by-two linear system with distinct pivot blocks. A block approximate factorization preconditioning method that is well-suited for approximating the Schur complement is considered by utilizing the so called matching strategy. A distinctive feature of the proposed preconditioner is its computational efficiency arising from its practical Schur complement-free implementation manner. Furthermore, the eigenvalues of the preconditioned system are demonstrated to lie within parameter-free positive real intervals, ensuring fast convergence independent of problem parameters under Krylov subspace acceleration. Motivated by the inherent block-Toeplitz structures, circulant-based inexact variants of the proposed preconditioner are explored and implemented within diagonalization strategies by fast Fourier transformation (FFT). Numerical experiments are conducted to validate the effectiveness and robustness of our proposed preconditioners compared with some optimal preconditioning strategies.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117348"},"PeriodicalIF":2.6,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145982028","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}
Pub Date : 2026-09-01Epub Date: 2026-01-13DOI: 10.1016/j.cam.2026.117362
Narayanaswamy Balakrishnan , María Jaenada , Leandro Pardo
Many modern devices are highly reliable, with long lifetimes before their failure. Conducting reliability tests under actual use conditions may require therefore impractically long experimental times to gather sufficient data for developing accurate inference. To address this, Accelerated Life Tests (ALTs) are often used in industrial experiments to induce product degradation and eventual failure more quickly by increasing certain environmental stress factors. Data collected under such increased stress conditions are analyzed, and results are then extrapolated to normal operating conditions. These tests typically involve a small number of devices and so pose significant challenges, such as interval-censoring. As a result, the outcomes are particularly sensitive to outliers in the data. Additionally, a comprehensive analysis requires more than just point estimation; inferential methods such as confidence intervals and hypothesis testing are essential to fully assess the reliability behaviour of the product.
This paper presents robust statistical methods based on minimum divergence estimators for analyzing ALT data of highly reliable devices under step-stress conditions and Gamma lifetime distributions. Robust test statistics generalizing the Rao test and divergence-based tests for testing linear null hypothesis are then developed. These hypotheses include in particular tests for the significance of the identified stress factors and for the validity of the assumption of exponential lifetimes.
{"title":"Robust divergence-based tests of hypotheses for simple step-stress accelerated life-testing under Gamma lifetime distributions","authors":"Narayanaswamy Balakrishnan , María Jaenada , Leandro Pardo","doi":"10.1016/j.cam.2026.117362","DOIUrl":"10.1016/j.cam.2026.117362","url":null,"abstract":"<div><div>Many modern devices are highly reliable, with long lifetimes before their failure. Conducting reliability tests under actual use conditions may require therefore impractically long experimental times to gather sufficient data for developing accurate inference. To address this, Accelerated Life Tests (ALTs) are often used in industrial experiments to induce product degradation and eventual failure more quickly by increasing certain environmental stress factors. Data collected under such increased stress conditions are analyzed, and results are then extrapolated to normal operating conditions. These tests typically involve a small number of devices and so pose significant challenges, such as interval-censoring. As a result, the outcomes are particularly sensitive to outliers in the data. Additionally, a comprehensive analysis requires more than just point estimation; inferential methods such as confidence intervals and hypothesis testing are essential to fully assess the reliability behaviour of the product.</div><div>This paper presents robust statistical methods based on minimum divergence estimators for analyzing ALT data of highly reliable devices under step-stress conditions and Gamma lifetime distributions. Robust test statistics generalizing the Rao test and divergence-based tests for testing linear null hypothesis are then developed. These hypotheses include in particular tests for the significance of the identified stress factors and for the validity of the assumption of exponential lifetimes.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117362"},"PeriodicalIF":2.6,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039551","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}
Battery cycling, both in application and Research and Development (R&D) environments, generates a wealth of information that often remains underexploited. Thus, potentially valuable information contained in electrical transients is frequently overlooked. In this framework, battery response modelling and model-based data analysis provide powerful tools to extract valuable information on battery status, its evolution and its correlation with functional performance. In this scenario, recently, we have developed a PDE model of battery potential response controlled by electrode shape changes in the BCs for high energy-density metal electrodes. In this work, on the basis of this model, we carry out a classification of the potential transient types, to enable a systematic comparison between model solution and experimental time-series. For transient shape classification purposes, we found that cluster analysis can play a key role in discovering hidden structures within the data. Specifically, in this paper, we apply the K-Means clustering algorithm to classify voltage profiles obtained as numerical solutions of the PDE model for the case of symmetric Li/Li cells. We introduce a weighted discrete Sobolev distance that allows us to spot changes in the shape of the voltage profiles, such as formation of peaks, valleys and concavities, that standard metrics such as the norm fail to capture. As an application, we consider a selection of experimental galvanostatic discharge-charge potential time-series to classify their shape in terms of cluster centroids. Moreover, we show that the new clustering algorithm can provide a segmentation of the parameter space of the PDE model. This partitioning is useful to link the experimental profiles to specific parameter ranges. In particular, we report an example to validate the fitting results of a recent publication of ours obtained via a Deep Learning approach for the same measured profiles.
{"title":"Shape classification of battery cycling profiles via K-Means clustering based on a Sobolev distance","authors":"Maria Grazia Quarta , Ivonne Sgura , Massimo Frittelli , Raquel Barreira , Benedetto Bozzini","doi":"10.1016/j.cam.2026.117365","DOIUrl":"10.1016/j.cam.2026.117365","url":null,"abstract":"<div><div>Battery cycling, both in application and Research and Development (R&D) environments, generates a wealth of information that often remains underexploited. Thus, potentially valuable information contained in electrical transients is frequently overlooked. In this framework, battery response modelling and model-based data analysis provide powerful tools to extract valuable information on battery status, its evolution and its correlation with functional performance. In this scenario, recently, we have developed a PDE model of battery potential response controlled by electrode shape changes in the BCs for high energy-density metal electrodes. In this work, on the basis of this model, we carry out a classification of the potential transient types, to enable a systematic comparison between model solution and experimental time-series. For transient shape classification purposes, we found that cluster analysis can play a key role in discovering hidden structures within the data. Specifically, in this paper, we apply the K-Means clustering algorithm to classify voltage profiles obtained as numerical solutions of the PDE model for the case of symmetric Li/Li cells. We introduce a <em>weighted discrete Sobolev distance</em> that allows us to spot changes in the shape of the voltage profiles, such as formation of peaks, valleys and concavities, that standard metrics such as the <span><math><msup><mrow><mi>L</mi></mrow><mn>2</mn></msup></math></span> norm fail to capture. As an application, we consider a selection of experimental galvanostatic discharge-charge potential time-series to classify their shape in terms of cluster centroids. Moreover, we show that the new clustering algorithm can provide a segmentation of the parameter space of the PDE model. This partitioning is useful to link the experimental profiles to specific parameter ranges. In particular, we report an example to validate the fitting results of a recent publication of ours obtained via a Deep Learning approach for the same measured profiles.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117365"},"PeriodicalIF":2.6,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039581","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}
Pub Date : 2026-09-01Epub Date: 2026-01-06DOI: 10.1016/j.cam.2026.117350
Xue-Lei Lin , Chengyu Chen , Ye Liu , Huifang Yuan
The alternating direction implicit (ADI) scheme is a type of well-known fast solvable scheme, which is frequently studied for time-dependent problems with constant coefficients on rectangular domains. Due to the complicated algebraic matrix structure, it is rare to study the ADI schemes for time-dependent problems with variable coefficients on irregular domains in the literature. In this paper, we study a novel ADI scheme for the well-known challenging problem: convection-dominated convection diffusion equation with variable coefficients on two-dimensional irregular domain. An upwind difference scheme is proposed for the spatial discretization, which leads to diagonally dominant spatial discretization matrices. The ADI temporal discretization reduces the two-dimension spatial problem into two one-dimension spatial sub-problems, in which the sub-problem along x-direction has a tridiagonal structure while the sub-problem along y-direction is sparse and unstructured due to the complicated domain geometry. To handle the complicated structure of the y-sub-problems, a grid adaptive permutation technique is proposed to convert the y-sub-problems into tridiagonal systems. As a result, all the one-dimension sub-problems arising from the proposed ADI schemes are diagonally dominant tridiagonal systems, which can be fast and directly solved by banded LU factorization with optimal complexity (i.e., linear complexity). It is well-known that the complicated domain geometry and the dominance of the convection terms bring challenge to numerical solution of the equation. Remarkably, both theoretical results and numerical results show that the proposed scheme is unconditionally stable with respect to the dominance of the convection term, time-space grid ratio and is flexible to the domain geometry.
{"title":"An ADI scheme for convection diffusion equation with variable coefficients on irregular domain","authors":"Xue-Lei Lin , Chengyu Chen , Ye Liu , Huifang Yuan","doi":"10.1016/j.cam.2026.117350","DOIUrl":"10.1016/j.cam.2026.117350","url":null,"abstract":"<div><div>The alternating direction implicit (ADI) scheme is a type of well-known fast solvable scheme, which is frequently studied for time-dependent problems with constant coefficients on rectangular domains. Due to the complicated algebraic matrix structure, it is rare to study the ADI schemes for time-dependent problems with variable coefficients on irregular domains in the literature. In this paper, we study a novel ADI scheme for the well-known challenging problem: convection-dominated convection diffusion equation with variable coefficients on two-dimensional irregular domain. An upwind difference scheme is proposed for the spatial discretization, which leads to diagonally dominant spatial discretization matrices. The ADI temporal discretization reduces the two-dimension spatial problem into two one-dimension spatial sub-problems, in which the sub-problem along <em>x</em>-direction has a tridiagonal structure while the sub-problem along <em>y</em>-direction is sparse and unstructured due to the complicated domain geometry. To handle the complicated structure of the <em>y</em>-sub-problems, a grid adaptive permutation technique is proposed to convert the <em>y</em>-sub-problems into tridiagonal systems. As a result, all the one-dimension sub-problems arising from the proposed ADI schemes are diagonally dominant tridiagonal systems, which can be fast and directly solved by banded LU factorization with optimal complexity (i.e., linear complexity). It is well-known that the complicated domain geometry and the dominance of the convection terms bring challenge to numerical solution of the equation. Remarkably, both theoretical results and numerical results show that the proposed scheme is unconditionally stable with respect to the dominance of the convection term, time-space grid ratio and is flexible to the domain geometry.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117350"},"PeriodicalIF":2.6,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145982029","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}
Pub Date : 2026-09-01Epub Date: 2026-01-18DOI: 10.1016/j.cam.2026.117372
Jana Vráblíková, Bert Jüttler
Computing the envelope of deforming planar domains is a significant and challenging problem with a wide range of potential applications. We approximate the envelope using circular arc splines, curves that balance geometric flexibility and computational simplicity. Our approach combines two concepts to achieve these benefits.
First, we represent a planar domain by its medial axis transform (MAT), which is a geometric graph in Minkowski space (possibly with degenerate branches). We observe that circular arcs in the Minkowski space correspond to MATs of arc spline domains. Furthermore, as a planar domain evolves over time, each branch of its MAT evolves and forms a surface in the Minkowski space. This allows us to reformulate the problem of envelope computation as a problem of computing cyclographic images of finite sets of curves on these surfaces. We propose and compare two pairs of methods for approximating the curves and boundaries of their cyclographic images. All of these methods result in an arc spline approximation of the envelope of the evolving domain.
Second, we exploit the geometric flexibility of circular arcs in both the plane and Minkowski space to achieve a high approximation rate. The computational simplicity ensures the efficient trimming of redundant branches of the generated envelope using a sweep line algorithm with optimal computational complexity.
{"title":"Arc spline approximation of envelopes of evolving planar domains","authors":"Jana Vráblíková, Bert Jüttler","doi":"10.1016/j.cam.2026.117372","DOIUrl":"10.1016/j.cam.2026.117372","url":null,"abstract":"<div><div>Computing the envelope of deforming planar domains is a significant and challenging problem with a wide range of potential applications. We approximate the envelope using circular arc splines, curves that balance geometric flexibility and computational simplicity. Our approach combines two concepts to achieve these benefits.</div><div>First, we represent a planar domain by its medial axis transform (MAT), which is a geometric graph in Minkowski space <span><math><msup><mi>R</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span> (possibly with degenerate branches). We observe that circular arcs in the Minkowski space correspond to MATs of arc spline domains. Furthermore, as a planar domain evolves over time, each branch of its MAT evolves and forms a surface in the Minkowski space. This allows us to reformulate the problem of envelope computation as a problem of computing cyclographic images of finite sets of curves on these surfaces. We propose and compare two pairs of methods for approximating the curves and boundaries of their cyclographic images. All of these methods result in an arc spline approximation of the envelope of the evolving domain.</div><div>Second, we exploit the geometric flexibility of circular arcs in both the plane and Minkowski space to achieve a high approximation rate. The computational simplicity ensures the efficient trimming of redundant branches of the generated envelope using a sweep line algorithm with optimal computational complexity.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117372"},"PeriodicalIF":2.6,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039576","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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