Article
Mathematics, Applied
Maria Luz Alvarez, Ricardo G. Duran
Summary: This article discusses the application of the Raviart-Thomas mixed finite element method to non-uniform elliptic problems. It introduces an error estimator based on local post-processing and proves its efficiency and reliability, generalizing the theory developed in [24] to degenerate cases. Finally, the authors present numerical computations demonstrating the good performance of an adaptive procedure based on their estimator.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Ying Liu, Yufeng Nie
Summary: This paper derives the a priori and a posteriori error estimates for the weak Galerkin finite element method with CrankNicolson time discretization applied to parabolic equations. The a priori estimates are based on existing results for elliptic projection problems, while the a posteriori estimates use an elliptic reconstruction technique to decompose the true error into elliptic and parabolic components. These estimates are further used to develop a temporal and spatial adaptive algorithm, with numerical results provided to validate the proposed estimators on uniform and adaptive meshes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Mahboub Baccouch
Summary: This paper derives two a posteriori error estimates for the LDG method applied to linear second-order elliptic problems on Cartesian grids, showing the superconvergence of the LDG solution gradient and introducing a postprocessing gradient recovery scheme. The proposed residual-type and recovery-type a posteriori error estimates converge to the true errors in the L-2-norm under mesh refinement with p + 1 order of convergence. Additionally, a local adaptive mesh refinement procedure is presented based on local and global a posteriori error estimates, validated through numerical examples.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Shipeng Xu
Summary: This paper derives a posteriori error estimates for the Weak Galerkin finite element methods for second order elliptic problems in terms of an H1-equivalent energy norm. The error analysis of the methods is proven to be valid for polygonal meshes under general assumptions, making it possible to solve Stokes equations and biharmonic equations on such meshes. The theoretical findings are verified by numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Computer Science, Artificial Intelligence
Nicolas Barnafi, Gabriel N. Gatica, Daniel E. Hurtado, Willian Miranda, Ricardo Ruiz-Baier
Summary: Deformable image registration (DIR) is a popular technique for aligning digital images, especially in medical image analysis. This study proposes adaptive mesh refinement schemes for the finite-element solution of DIR problems, which have shown to significantly reduce the number of degrees of freedom without compromising solution accuracy. The adaptive scheme performs well in numerical convergence on smooth synthetic images and successfully handles volume-constrained registration problems.
SIAM JOURNAL ON IMAGING SCIENCES
(2021)
Article
Mathematics, Applied
Stefan Kurz, Dirk Pauly, Dirk Praetorius, Sergey Repin, Daniel Sebastian
Summary: Functional error estimates are well-established tools for a posteriori error estimation and adaptive mesh-refinement in the finite element method (FEM). This work proposes a functional error estimate for the boundary element method (BEM) that provides guaranteed lower and upper bounds for the unknown error. The analysis covers Galerkin BEM and collocation method, making it of particular interest for scientific computations and engineering applications.
NUMERISCHE MATHEMATIK
(2021)
Article
Mathematics, Applied
Mahboub Baccouch
Summary: This paper investigates the superconvergence properties of the local discontinuous Galerkin method for linear second-order elliptic problems on Cartesian grids, and presents an efficient and reliable a posteriori error estimator. The proposed method converges to the true errors under mesh refinement, with a convergence order of p + 2.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
G. M. M. Reddy, P. Nanda, M. Vynnycky, J. A. Cuminato
Summary: This article introduces a novel computational technique for efficiently solving the inverse boundary identification problem with uncertain data in two dimensions. The method relies on a posteriori error indicators using Tikhonov regularized solutions obtained by the method of fundamental solutions (MFS) and given data. An adaptive stochastic optimization strategy is used for stable solutions, avoiding unstable regions.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Yuanyuan Zhang, Xiaoping Liu
Summary: This article presents a residual-based a posteriori error estimator for a quadratic finite volume method for solving nonlinear elliptic partial differential equations. It is shown that the estimator provides global upper and local lower bounds for the H1 norm error of the method. Numerical experiments confirm the theoretical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Andrea Cangiani, Emmanuil H. Georgoulis, Oliver J. Sutton
Summary: This article presents posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. The practical interest of this setting is demonstrated by applying the results to finite element methods on moving meshes and using estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The error estimates, derived using the elliptic reconstruction technique in an abstract framework, significantly relax the basic assumption underlying previous estimates by not requiring any particular compatibility between computational meshes used on consecutive time-steps.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2021)
Article
Engineering, Electrical & Electronic
Jake J. Harmon, Cam Key, Donald Estep, Troy Butler, Branislav M. Notaros
Summary: The paper presents the application of adjoint analysis in 3-D finite-element method scattering problems for a posteriori error estimation and adaptive refinement, which can significantly improve efficiency and accuracy in computational electromagnetics (CEM). The formulation of adjoint problem of the 3-D double-curl wave equation and error estimates for novel accelerated adaptive refinement algorithms are demonstrated. The proposed refinement algorithms enable rapid refinement from coarse initial discretizations to high accuracy, reducing manual intervention and accelerating automatic refinement to fine tolerances.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
(2021)
Article
Computer Science, Interdisciplinary Applications
Assyr Abdulle, Giacomo Rosilho de Souza
Summary: This paper introduces a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The method improves the accuracy of the solution by solving local elliptic problems in refined subdomains and provides an algorithm for the automatic identification of these subdomains. Numerical comparisons demonstrate the efficiency of the method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Ying Wang, Gang Wang, Feng Wang
Summary: This paper presents and analyzes a residual-type a posteriori error estimator for low-order virtual element discretization for the Stokes and Navier-Stokes problems, proving its globally upper and locally lower bounds for discretization error, with modifications for small viscosity cases. The effectiveness and flexibility of the designed error estimator combined with adaptive mesh refinement are verified through a series of benchmark tests.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Stefano Giani, Luka Grubisic, Harri Hakula, Jeffrey S. Ovall
Summary: The study introduces an a posteriori error estimator for high-order p- or hp-finite element discretizations of selfadjoint linear elliptic eigenvalue problems, which is effective in estimating the approximation error in eigenvalue clusters and their corresponding invariant subspaces.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Ram Manohar, Rajen Kumar Sinha
Summary: In this study, a posteriori error estimates of the finite-element method for linear parabolic optimal control problems in a convex bounded polyhedral domain are derived. Variational discretization is used to approximate the state, co-state, and control variables, while temporal discretization is based on the backward Euler method. The error analysis relies on the elliptic reconstruction technique and heat kernel estimates, resulting in posterior error estimates for the state, co-state, and control variables in the L-infinity(0, T; L-infinity(Omega))-norm.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Lin Mu, Xu Zhang
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Lin Mu
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
James H. Adler, Xiaozhe Hu, Lin Mu, Xiu Ye
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Feng Bao, Lin Mu, Jin Wang
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2019)
Article
Engineering, Multidisciplinary
Gang Wang, Lin Mu, Ying Wang, Yinnian He
Summary: This paper introduces a pressure-robust virtual element method for solving the Stokes problem on convex polygonal meshes. By enhancing the approximation methods for velocity and pressure, pressure-independent velocity approximation is achieved, with numerical experiments validating the theoretical conclusions.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Lin Mu, Xiu Ye, Shangyou Zhang
Summary: Pressure-robustness is crucial for incompressible fluid simulations. Enhancements to the discontinuous Galerkin finite element methods in the primary velocity-pressure formulation for solving Stokes equations have been developed to achieve pressure-robustness. The new schemes show improvements in source term modifications and have been validated through numerical experiments. Optimal-order error estimates have been established for the numerical approximations in various norms.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Bin Wang, Ingo Wald, Nate Morrical, Will Usher, Lin Mu, Karsten Thompson, Richard Hughes
Summary: A novel efficient and robust particle tracking method (RT method) is presented to accelerate Eulerian-Lagrangian simulations using hardware ray tracing cores and GPU parallel computing technology. The method includes a hardware-accelerated hosting cell locator and a robust treatment of particle-wall interaction, demonstrated through numerical simulations and experimental observations. Benchmark results show a significant performance improvement compared to the reference method for large-scale simulations.
COMPUTER PHYSICS COMMUNICATIONS
(2022)
Article
Mathematics, Applied
Guannan Zhang, Lin Mu
Summary: A non-intrusive domain-decomposition model reduction method has been developed for linear steady-state PDEs with random-field coefficients, enabling the construction of a reduced model without intrusive implementation from scratch by accessing the final linear system of a deterministic PDE solver.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
David Green, Xiaozhe Hu, Jeremy Lore, Lin Mu, Mark L. Stowell
Summary: In this paper, an interior penalty discontinuous Galerkin finite element scheme is presented for solving diffusion problems with strong anisotropy arising in magnetized plasmas for fusion applications. The authors demonstrate the accuracy of the high-order scheme and develop an efficient preconditioning technique that is robust to the mesh size and anisotropy of the problem. Several numerical tests are provided to validate the accuracy and efficiency of the proposed algorithm.
COMPUTER PHYSICS COMMUNICATIONS
(2022)
Article
Mathematics, Applied
Lin Mu
Summary: In this article, a novel numerical scheme for solving the steady incompressible Navier-Stokes equations is developed and analyzed using the weak Galerkin methods. The algorithm achieves pressure-robustness by employing a divergence-preserving velocity reconstruction operator, which ensures that the velocity error is independent of the pressure and irrotational body force. Error analysis is conducted to establish the convergence rate, and numerical experiments are presented to validate the theoretical conclusions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Lin Mu, Xiu Ye, Shangyou Zhang
Summary: This paper proposes a new stabilizer-free and pressure-robust weak Galerkin finite element method for the Stokes equations with superconvergence, which is validated through numerical experiments for its effectiveness and robustness.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Eduardo D'Azevedo, David L. Green, Lin Mu
COMPUTER PHYSICS COMMUNICATIONS
(2020)
Article
Mathematics, Applied
Lin Mu
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Lin Mu
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2020)
Article
Mathematics, Applied
Lin Mu, Guannan Zhang
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2019)
Article
Mathematics, Applied
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
