Article
Computer Science, Interdisciplinary Applications
Will Thacher, Hans Johansen, Daniel Martin
Summary: We propose a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. The method demonstrates second, fourth, and sixth order accuracy on various tests, including problems with high contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We develop a generalized truncation error analysis and a simple method based on Green's theorem for computing exact geometric moments, which enable easy inclusion of spatially-varying coefficients and jump conditions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics
Christian Baer, Lashi Bandara
Summary: We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. We demonstrate the equivalence of various characterisations of elliptic boundary conditions and prove the regularity of solutions up to the boundary. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact, and provide examples treated conveniently by our methods.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Zeinab Gharibi, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: In this study, the authors developed an immersed weak Galerkin finite element method (IWG-FEM) for solving elliptic interface problems. It is proven that the method has optimal convergence in both energy norm and L-2 norm under natural smoothness assumption on the solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
H. H. Kim, C-Y Jung, T. B. Nguyen
Summary: This article presents a staggered discontinuous Galerkin (SDG) approximation for elliptic problems on rectangular meshes, with theoretical proofs of optimal convergence results. Numerical evidence and examples demonstrate the effectiveness, stability, and accuracy of the proposed method. The simplicity of rectangular meshes allows for easy definition of discrete gradients, making numerical implementation easier, and there are plans to extend this approach to curved domains in future research.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Physics, Mathematical
Minhyun Kim, Ki-Ahm Lee, Se-Chan Lee
Summary: This paper investigates the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability s ∈ (0, 1) and summability p > 1. It establishes a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Mathematics
Nestor F. Castaneda-Centurion, Lucas C. F. Ferreira
Summary: In this study, a class of elliptic problems in the half-space R-+(n) with nonhomogeneous boundary conditions containing nonlinearities and critical singular potentials are considered. The existence and regularity results are obtained through harmonic analysis approach based on a framework of weighted spaces in Fourier variables. The results cover supercritical nonlinearities and include various versions of Kato's and Hardy's potentials without using Kato and Hardy inequalities.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Quanxiang Wang, Zhiyue Zhang, Liqun Wang
Summary: A new immersed finite volume element method is proposed in this paper to solve elliptic problems on Cartesian mesh with discontinuous diffusion coefficient and sharp-edged interfaces. Extensive numerical experiments show that the method achieves approximately second-order convergence for piecewise smooth solutions, and more than 1.65th order accuracy for solutions with singularity.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Yue Wang, Fuzheng Gao, Jintao Cui
Summary: A new conforming discontinuous Galerkin method is proposed for solving second order elliptic interface problems with discontinuous coefficient. Compared with known weak Galerkin algorithms, the method studied in this paper has no stabilizer and fewer unknowns. Error estimates in H-1 and L-2 norms are established, showing optimal order convergence.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Gule Rana, Muhammad Asif, Nadeem Haider, Rubi Bilal, Muhammad Ahsan, Qasem Al-Mdallal, Fahd Jarad
Summary: A new numerical technique for the simulations of advection-diffusion-reaction type elliptic and parabolic interface models is proposed, which combines the Haar wavelet collocation method and the finite difference method. The technique shows more efficiency, better accuracy, and simpler applicability compared to existing methods, as demonstrated by the obtained numerical results.
JOURNAL OF FUNCTION SPACES
(2022)
Article
Computer Science, Interdisciplinary Applications
Kejia Pan, Xiaoxin Wu, Yufeng Xu, Guangwei Yuan
Summary: This paper proposes an easy-to-implement exact-interface-fitted mesh generation algorithm and a linearity preserving finite volume scheme for solving anisotropic elliptic interface problems with non-homogeneous jump conditions. The algorithm generates a structured exact-interface-fitted curved quadrilateral mesh, and the derived finite volume scheme can achieve linearity preservation on curved quadrilateral meshes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
S. J. Castillo, J. Karatson
Summary: This paper focuses on the iterative solution of finite element discretizations for second-order elliptic boundary value problems. Mesh independent estimations are provided for the rate of superlinear convergence of preconditioned Krylov methods, exploring the connection between the convergence rate and the Lebesgue exponent of the data. Numerical examples are presented to demonstrate the theoretical results.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics
Luca Di Fazio, Emanuele Spadaro
Summary: Variational inequalities with thin obstacles and Signorini-type boundary conditions are classical problems in the calculus of variations. While refined results are known in the linear case, our understanding in the nonlinear setting is still preliminary. In this paper, we prove C-1 regularity for solutions to a general class of quasi-linear variational inequalities with thin obstacles, and C-1, C-alpha regularity for variational inequalities under Signorini-type conditions on the boundary of a domain.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Sunhi Choi, Inwon C. Kim
Summary: We consider a nonlinear Neumann problem with periodic oscillation in the elliptic operator and on the boundary condition. We focus on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of oscillation increases, we derive quantitative homogenization results. When the homogenized operator is rotation-invariant, we prove the Holder continuity of the homogenized boundary data. Our results appear to be new even for the linear oblique problem, and we improve and optimize the rate of convergence within our approach.
ADVANCED NONLINEAR STUDIES
(2023)
Article
Mathematics
G. Metafune, L. Negro, C. Spina
Summary: In this study, we investigate elliptic and parabolic problems governed by the singular elliptic operators (I L=y alpha 1Ax+y alpha 2Dyy+ycDy-b, α1, α2∈R, y2 in the half-space RN+1+={(x, y) : x∈RN, y>0}. We prove Lp-estimates and solvability for the associated problems. Using semigroup theory, we show that L generates an analytic semigroup, characterize its domain as a weighted Sobolev space, and demonstrate its maximal regularity. (c) 2022 Elsevier Inc. All rights reserved.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Waixiang Cao, Chunmei Wang, Junping Wang
Summary: This article introduces a new primal-dual weak Galerkin (PDWG) finite element method for second order elliptic interface problems with ultra-low regularity assumptions. It is proved that the PDWG method is stable and accurate with optimal order of error estimates. Extensive numerical experiments verify the efficiency and accuracy of the new PDWG method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
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
Mathematics, Applied
Lin Mu
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(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
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)