Article
Mathematics, Applied
Stefan Sauter
Summary: In this paper, the discretization of the two-dimensional stationary Stokes equation using CrouzeixRaviart elements for the velocity of polynomial order k >= 1 and discontinuous pressure approximations of order k - 1 is considered. The lower bound of the inf-sup constant is bounded independently of the mesh size and is shown to depend only logarithmically on k. The assumptions on the mesh are very mild: at least one inner vertex for odd k and more than a single triangle for even k.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Srinath Mahankali, Yunan Yang
Summary: This study analyzes the forward and inverse scattering series for scalar waves based on the Helmholtz equation and the diffuse waves from the time-independent diffusion equation. The stability and approximation error of the series under the Sobolev H (s) norm are studied, and comparisons are made using different H (s) norms for both the parameter and data domains. The theoretical justification for frequency weighting techniques in practical inversion procedures is provided, along with numerical inversion examples.
Article
Mathematics, Applied
Maximilian Bernkopf, Stefan Sauter, Celine Torres, Alexander Veit
Summary: This paper focuses on studying the unique solvability of the discretized Helmholtz problem with Robin boundary conditions using a conforming Galerkin finite element method. By mimicking the tools for proving well-posedness of the continuous problem directly on the discrete level, a computable criterion is derived to certify discrete well-posedness.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Jiayuan Liu, Tingting Wu, Taishan Zeng
Summary: This paper provides a numerical analysis of the optimal 9-point finite difference format for the two-dimensional Helmholtz equation with constant wavenumber. The approach involves transforming the 2D error equation into a series of 1D difference equations and proving the existence, uniqueness, and stability of the solutions for each 1D difference equation. Based on the results of the one-dimensional problems, the uniqueness and convergence of the solution for the optimal 9-point scheme are derived. Numerical experiments confirm that the optimal 9-point difference scheme has second-order convergence.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Ivy Weber, Gunilla Kreiss, Murtazo Nazarov
Summary: This paper investigates the stability of a numerical method for solving the wave equation using matrix eigenvalue analysis to calculate time-step restrictions. It is found that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, while the restriction for symmetric interior penalty DG schemes is tighter. The best time-step restriction is obtained for continuous Hermite finite elements.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Yvon Maday, Carlo Marcati
Summary: We study a class of nonlinear eigenvalue problems with singular potentials, which are commonly found in physics and chemistry. The results show that, under certain conditions, the eigenfunctions belong to analytic-type non-homogeneous weighted Sobolev spaces. By approximating the solution using an isotropically refined hp method, the numerical solution converges exponentially to the exact eigenfunction.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2023)
Article
Engineering, Multidisciplinary
Denis Duhamel
Summary: This paper presents a method for calculating high-frequency wave radiation in exterior domains using finite element methods. By decomposing the solution into an analytical part and a numerical part in a series expansion, the solution to the radiation or scattering problem can be obtained by solving a sparse linear system. The proposed method offers advantages in terms of computational efficiency and result accuracy.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Suleyman Cengizci, Omur Ugur
Summary: This computational study investigates numerical solutions of Burgers'-type equations at high Reynolds numbers. The nonlinearity becomes significant and the equations are dominated by convection, leading to spurious oscillations in solutions obtained through standard numerical methods. To address this, the Galerkin finite element formulation is stabilized using the streamline-upwind/Petrov-Galerkin method. YZ beta shock-capturing is further employed to improve solution profiles around strong gradients. The resulting nonlinear equation systems are solved using the Newton-Raphson (N-R) method, while the linearized equation systems are solved with the BiCGStab technique and ILU preconditioning.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Peijun Li, Jian Zhai, Yue Zhao
Summary: This paper addresses an inverse source problem for the three-dimensional Helmholtz equation, showing uniqueness and Lipschitz-type stability estimate under the assumption that the source function is piecewise constant on a domain made up of a union of disjoint convex polyhedral subdomains.
Article
Mathematics, Applied
E. A. Spence
Summary: In d dimensions, accurately approximating an arbitrary function with frequency <= k requires approximately k(d) degrees of freedom. The pollution effect occurs in a numerical method for solving the Helmholtz equation if the total number of degrees of freedom needed to maintain accuracy grows faster than the natural threshold as k approaches infinity. The hp-FEM, which increases accuracy by decreasing the meshwidth h and increasing the polynomial degree p, does not suffer from the pollution effect unlike the h-version of the finite element method.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Axel Modave, Theophile Chaumont-Frelet
Summary: In this paper, a new hybridizable discontinuous Galerkin (HDG) method called the CHDG method is proposed for solving time-harmonic scalar wave propagation problems. The method utilizes a standard discontinuous Galerkin scheme with upwind numerical fluxes and high-order polynomial bases. Auxiliary unknowns are introduced at the element interfaces to simplify the system, which can be solved using stationary iterative schemes. Numerical results show that the CHDG method improves the properties of the reduced system compared to the standard HDG method and is more suitable for iterative solution procedures.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Shujuan Lu, Tao Xu, Zhaosheng Feng
Summary: In this study, a second-order finite difference scheme is proposed for analyzing a class of space-time variable-order fractional diffusion equation. The scheme is demonstrated to be unconditionally stable and convergent with a convergence order of O(tau(2) + h(2)) under certain conditions, as validated by numerical examples.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Tao Cui, Ziming Wang, Xueshuang Xiang
Summary: In this paper, a plane wave activation based neural network (PWNN) is proposed to efficiently solve the Helmholtz equation with constant coefficients and relatively large wave number kappa. PWNN significantly improves the computational speed and accuracy, compared to traditional activation based neural networks (TANN) and the finite difference method (FDM), especially for large wave number problems. Theoretical guidance is given based on new error estimates for choosing the number of neural network's neurons and the initial value to accelerate network training. Numerical experiments in 2D and 3D demonstrate the efficiency and accuracy of PWNN, particularly for large wave number problems.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Wim A. Mulder
Summary: This article discusses the importance of mass lumping method in solving the wave equation with finite elements. By augmenting higher-degree polynomials in the interior to maintain accuracy, various elements with positive weights have been found. Numerical tests on the new elements for wave propagation problem confirm their accuracy and computational efficiency.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Edward Laughton, Vidhi Zala, Akil Narayan, Robert M. Kirby, David Moxey
Summary: As the use of high-order finite element methods continues to expand, efficient algorithms associated with high-order operations receive increasing attention. This work focuses on solution expansion evaluation at arbitrary points within an element and presents efficient algorithms along with benchmarking results.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)