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Mathematics, Applied
Zhuoran Wang, Simon Tavener, Jiangguo Liu
Summary: This paper introduces a novel 2-field finite element solver for linear poroelasticity on convex quadrilateral meshes. It discretizes Darcy flow for fluid pressure and linear elasticity for solid displacement, coupling them through implicit Euler temporal discretization to solve poroelasticity problems. The rigorous error analysis and numerical tests demonstrate the accuracy and locking-free property of this new solver.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Ruishu Wang, Zhuoran Wang, Jiangguo Liu
Summary: This paper presents a family of new weak Galerkin finite element methods for solving linear elasticity in the primal formulation. These methods use vector-valued polynomials of degree k >= 0 to approximate the displacement independently in element interiors and on edges of a convex quadrilateral mesh. The new methods do not require penalty or stabilizer and are free of Poisson-locking, while achieving optimal order (k + 1) convergence rates in displacement, stress, and dilation. Numerical experiments on popular test cases demonstrate the theoretical estimates and efficiency of these new solvers. The extension to cuboidal hexahedral meshes is briefly discussed.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Yue Wang, Fuzheng Gao, Jintao Cui
Summary: The lowest-order weak Galerkin finite element method is introduced and analyzed for second order elliptic interface problems with discontinuous coefficients and solutions. It has a simpler formulation and superconvergence property compared to existing methods for interface problems, making it suitable for more complicated interfaces and domain configurations. The theoretical results are verified through numerical experiments, achieving optimal convergence rates for examples with low-regularity solutions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Giovanni Taraschi, Maicon R. Correa
Summary: In this paper, the Primal Hybrid Finite Element Method is used to approximate a second order elliptic equation (Darcy problem) on quadrilateral meshes. New results are presented regarding the sufficient and, in some cases, necessary conditions to achieve optimal convergence rates on convex quadrilaterals obtained from bilinear mappings. Numerical experiments are conducted to demonstrate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
T. M. Bendall, G. A. Wimmer
Summary: This paper presents two schemes to improve the accuracy of transport of vector-valued fields on two-dimensional curved manifolds. The first scheme reconstructs the transported field in a higher-order function space and solves the transport equation. The second scheme applies a mixed finite element formulation and solves for the transported field and its vorticity simultaneously. Numerical tests demonstrate the accuracy and improvement of these schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Y. Pan, P-O Persson
Summary: This paper presents a novel approach for high-order accurate numerical differentiation on quadrilateral meshes. By defining an auxiliary function with greater smoothness properties and differentiating it, the derivatives of the original function can be obtained. The method can be applied to meshes of arbitrary topology in any number of dimensions for any order of derivative and accuracy.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Jijing Zhao, Hongxing Rui
Summary: In this paper, a method combining weak Galerkin and conforming finite element is proposed to solve the hybrid-dimensional fracture problem. The method is shown to be accurate and robust through numerical examples.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Gang Wang, Ying Wang, Yinnian He
Summary: This paper introduces a weak Galerkin finite element method based on H(div) virtual element for Darcy flow on polytopal meshes. The pressure is approximated by constant functions and their discrete weak gradients are calculated in local H(div) virtual element spaces. The method provides locally mass-conservative numerical velocity with continuous normal fluxes by a simple L-2 projection after obtaining the numerical pressure.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Engineering, Multidisciplinary
Maicon R. Correa, Giovanni Taraschi
Summary: In this work, a computational strategy based on H(div,12) is proposed for post-processing the flux vector of the approximated solution of a second-order elliptic equation. The recovery strategy utilizes ABFt spaces to achieve locally conservative approximations of the flux and has been proven to provide optimal order approximations on quadrilateral meshes.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Constantin Bacuta, Leszek Demkowicz, Jaime Mora, Christos Xenophontos
Summary: This work focuses on two problems: analyzing the DPG method in fractional energy spaces, and investigating a non-conforming version of the DPG method for general polyhedral meshes. The ultraweak variational formulation is used for the model Laplace equation, and theoretical estimates are supported by 3D numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Antti H. Niemi
Summary: We investigate the approximation properties of the primal discontinuous Petrov-Galerkin (DPG) method on quadrilateral meshes. Our study extends the previous convergence results and duality arguments to cover arbitrary convex quadrilateral elements with bilinear isomorphisms. The theoretical findings are validated by numerical experiments, which also compare the primal DPG method with a conventional least squares finite element method with the same number of degrees of freedom.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Yanhui Zhou, Jiming Wu
Summary: This paper introduces a new method that utilizes a bubble function for postprocessing to obtain a finite volume element solution that converges to the analytic solution in the two-dimensional anisotropic diffusion problem.
APPLIED MATHEMATICS LETTERS
(2022)
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
Mathematics, Applied
Qingguang Guan, Gillian Queisser, Wenju Zhao
Summary: This paper proposes a method for defining new basis functions on curved sides or faces of curvilinear elements. The method collects linearly independent traces of polynomials on the curved sides/faces to construct the basis functions. It can handle complex boundaries or interfaces and achieve high-order convergence rates.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Kaifang Liu, Peng Zhu
Summary: This paper presents an error analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity requirements. It introduces a new error equation and optimal a priori error estimates, and verifies the theoretical results through numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Jiangguo Liu, Graham Harper, Nolisa Malluwawadu, Simon Tavener
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Jinying Tan, Jiangguo Liu
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2020)
Article
Computer Science, Software Engineering
Jehanzeb H. Chaudhry, Donald Estep, Zachary Stevens, Simon J. Tavener
Summary: This study investigates the posterior error estimates of a quantity that cannot be represented as a linear functional in differential equations, providing two representations and estimating unknown terms through an adjoint-based approach. The combination of these representations yields accurate results.
BIT NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Jiangguo Liu, Simon Tavener, Zhuoran Wang
JOURNAL OF SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Graham Harper, Ruishu Wang, Jiangguo Liu, Simon Tavener, Ran Zhang
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2020)
Article
Engineering, Multidisciplinary
Graham Harper, Jiangguo Liu, Simon Tavener, Tim Wildey
Summary: This paper presents a finite element method for solving coupled Stokes-Darcy flow problems by combining classical Bernardi-Raugel finite elements and the recently developed Arbogast-Correa (AC) spaces on quadrilateral meshes. A novel weak Galerkin methodology is employed for discretization of the Darcy equation, with piecewise constant approximants used to approximate the Darcy pressure. Rigorous error analysis and numerical experiments demonstrate the stability and optimal-order accuracy of the method.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Zhuoran Wang, Simon Tavener, Jiangguo Liu
Summary: This paper introduces a novel 2-field finite element solver for linear poroelasticity on convex quadrilateral meshes. It discretizes Darcy flow for fluid pressure and linear elasticity for solid displacement, coupling them through implicit Euler temporal discretization to solve poroelasticity problems. The rigorous error analysis and numerical tests demonstrate the accuracy and locking-free property of this new solver.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Biology
Ben Lambert, David J. Gavaghan, Simon J. Tavener
Summary: Variation among cells in biological systems is common, and mathematical models can be used to investigate these differences. Researchers have introduced a computational sampling method called Contour Monte Carlo for estimating mathematical model parameters from snapshot distributions. This method is suitable for underdetermined systems and can help researchers explore cellular variation in biological systems.
JOURNAL OF THEORETICAL BIOLOGY
(2021)
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Computer Science, Software Engineering
Jehanzeb H. Chaudhry, Donald Estep, Simon J. Tavener
Summary: The study focuses on adjoint-based a posteriori error analysis for domain decomposition methods, decomposing numerical errors into different contributions and constructing a two-stage solution strategy based on this decomposition to efficiently reduce errors in the quantity of interest by adjusting the relative contributions.
BIT NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Zhuoran Wang, Ruishu Wang, Jiangguo Liu
Summary: This paper presents novel finite element solvers for solving Stokes flow, which are pressure-robust due to the use of a lifting operator. The solvers utilize weak Galerkin finite element schemes and local Arbogast-Correa or Arbogast-Tao spaces for construction of discrete weak gradients. The lifting operator helps in removing pressure dependence of velocity errors, ensuring the robustness of the solvers. The theoretical validation and numerical illustrations demonstrate the pressure robustness of the solvers. A comparison with the non-robust classical Taylor-Hood solver is also provided.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Ruishu Wang, Zhuoran Wang, Jiangguo Liu
Summary: This paper presents a family of new weak Galerkin finite element methods for solving linear elasticity in the primal formulation. These methods use vector-valued polynomials of degree k >= 0 to approximate the displacement independently in element interiors and on edges of a convex quadrilateral mesh. The new methods do not require penalty or stabilizer and are free of Poisson-locking, while achieving optimal order (k + 1) convergence rates in displacement, stress, and dilation. Numerical experiments on popular test cases demonstrate the theoretical estimates and efficiency of these new solvers. The extension to cuboidal hexahedral meshes is briefly discussed.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Ruishu Wang, Zhuoran Wang, Jiangguo Liu, Simon Tavener, Ran Zhang
Summary: This paper presents numerical methods for solving linear elasticity on simplicial meshes based on enrichment of Lagrangian bilinear/trilinear finite elements. The classical 1st order Bernardi-Raugel spaces are innovatively used for this purpose, with a projection to the elementwise constant space to handle dilation in the strain-div formulation. Mixed boundary conditions are considered for error estimates in energy-norm and L-2-norms of displacement and stress, demonstrating methods free of Poisson-locking.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2021)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
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Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
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Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
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Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
