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This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used. The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes.

This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit (pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly. Our approach consists in directly discretizing the problem by means of Gautschi-type integrators based on $\operatorname{sinc}$ matrix functions. The novelty contained here is that of using a suitable rational approximation formula for the $\operatorname{sinc}$ matrix function to apply a rational Krylov-like approximation method with suitable choices of poles. In particular, we discuss the application of the whole strategy to a finite element discretization of the wave equation.

In this paper, both semidiscrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal $L^2$ error estimates for the semidiscrete as well as the fully discrete approximations of the velocity and of the pressure are derived for realistically assumed conditions on the data. The main ingredient in the proof is the appropriate exploitation of the inverse of the penalized Stokes operator, negative norm estimates and time weighted estimates. Numerical examples are discussed at the end which conform our theoretical results.

In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. More precisely, we prove the existence of discrete solutions, intended in a suitable weak sense, for which we prove a quasi-best approximation estimate similar to Cea's lemma, a result commonly found in finite element methods. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method's effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. The key requirement for achieving the best approximation outcome is to ensure that the space for the test network satisfies a specific condition, known as the inf-sup condition, essentially requiring that the test network set is sufficiently large when compared to the trial space. The method's accuracy, however, is only determined by the space of the trial network. We also devise a pseudo-time XNODE neural network class for static PDE problems, yielding significantly faster convergence results than the classical DNN network.

This paper presents a pressure-robust enriched Galerkin (EG) method for the Brinkman equations with minimal degrees of freedom based on EG velocity and pressure spaces. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. We derive, analyze, and compare two EG methods in this paper: standard and robust methods. The standard method requires a mesh size to be less than a viscous parameter to produce stable and accurate velocity solutions, which is impractical in the Darcy regime. Therefore, we propose the pressure-robust method by utilizing a velocity reconstruction operator and replacing EG velocity functions with a reconstructed velocity. The robust method yields error estimates independent of a pressure term and shows uniform performance from the Stokes to Darcy regimes, preserving minimal degrees of freedom. We prove well-posedness and error estimates for both the standard and robust EG methods. We finally confirm theoretical results through numerical experiments with two- and three-dimensional examples and compare the methods' performance to support the need for the robust method.

In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for solving the Stokes equations, which is an enhanced version of the EG scheme for the Stokes problem proposed in [Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James H. Adler, An enriched Galerkin method for the Stokes equations, Computers and Mathematics with Applications, accepted, 2022]. The pressure-robustness is achieved by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. An a priori error analysis proves that the velocity error is independent of the pressure and viscosity. We also propose and analyze a perturbed version of our pressure-robust EG method that allows for the elimination of the degrees of freedom corresponding to the discontinuous component of the velocity vector via static condensation. The resulting method can be viewed as a stabilized $H^1$-conforming $\mathbb{P}_1$-$\mathbb{P}_0$ method. Further, we consider efficient block preconditioners whose performances are independent of the viscosity. The theoretical results are confirmed through various numerical experiments in two and three dimensions.

The Transposition Distance Problem (TDP) is a classical problem in genome rearrangements which seeks to determine the minimum number of transpositions needed to transform a linear chromosome into another represented by the permutations $\pi$ and $\sigma$, respectively. This paper focuses on the equivalent problem of Sorting By Transpositions (SBT), where $\sigma$ is the identity permutation $\iota$. Specifically, we investigate palisades, a family of permutations that are "hard" to sort, as they require numerous transpositions above the celebrated lower bound devised by Bafna and Pevzner. By determining the transposition distance of palisades, we were able to provide the exact transposition diameter for $3$-permutations (TD3), a special subset of the Symmetric Group $S_n$, essential for the study of approximate solutions for SBT using the simplification technique. The exact value for TD3 has remained unknown since Elias and Hartman showed an upper bound for it. Another consequence of determining the transposition distance of palisades is that, using as lower bound the one by Bafna and Pevzner, it is impossible to guarantee approximation ratios lower than $1.375$ when approximating SBT. This finding has significant implications for the study of SBT, as this problem has been subject of intense research efforts for the past 25 years.

Several physical problems modeled by second-order partial differential equations can be efficiently solved using mixed finite elements of the Raviart-Thomas family for N-simplexes, introduced in the seventies. In case Neumann conditions are prescribed on a curvilinear boundary, the normal component of the flux variable should preferably not take up values at nodes shifted to the boundary of the approximating polytope in the corresponding normal direction. This is because the method's accuracy downgrades, which was shown in \cite{FBRT}. In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this paper an alternative with straight-edged triangles for two-dimensional problems is proposed. The key point of this method is a Petrov-Galerkin formulation of the mixed problem, in which the test-flux space is a little different from the shape-flux space. After carrying out a well-posedness and stability analysis, error estimates of optimal order are proven.

We present an efficient matrix-free point spread function (PSF) method for approximating operators that have locally supported non-negative integral kernels. The method computes impulse responses of the operator at scattered points, and interpolates these impulse responses to approximate integral kernel entries. Impulse responses are computed by applying the operator to Dirac comb batches of point sources, which are chosen by solving an ellipsoid packing problem. Evaluation of kernel entries allows us to construct a hierarchical matrix (H-matrix) approximation of the operator. Further matrix computations are performed with H-matrix methods. We use the method to build preconditioners for the Hessian operator in two inverse problems governed by partial differential equations (PDEs): inversion for the basal friction coefficient in an ice sheet flow problem and for the initial condition in an advective-diffusive transport problem. While for many ill-posed inverse problems the Hessian of the data misfit term exhibits a low rank structure, and hence a low rank approximation is suitable, for many problems of practical interest the numerical rank of the Hessian is still large. But Hessian impulse responses typically become more local as the numerical rank increases, which benefits the PSF method. Numerical results reveal that the PSF preconditioner clusters the spectrum of the preconditioned Hessian near one, yielding roughly 5x-10x reductions in the required number of PDE solves, as compared to regularization preconditioning and no preconditioning. We also present a numerical study for the influence of various parameters (that control the shape of the impulse responses) on the effectiveness of the advection-diffusion Hessian approximation. The results show that the PSF-based preconditioners are able to form good approximations of high-rank Hessians using a small number of operator applications.

We present an efficient matrix-free geometric multigrid method for the elastic Helmholtz equation, and a suitable discretization. Many discretization methods had been considered in the literature for the Helmholtz equations, as well as many solvers and preconditioners, some of which are adapted for the elastic version of the equation. However, there is very little work considering the reciprocity of discretization and a solver. In this work, we aim to bridge this gap. By choosing an appropriate stencil for re-discretization of the equation on the coarse grid, we develop a multigrid method that can be easily implemented as matrix-free, relying on stencils rather than sparse matrices. This is crucial for efficient implementation on modern hardware. Using two-grid local Fourier analysis, we validate the compatibility of our discretization with our solver, and tune a choice of weights for the stencil for which the convergence rate of the multigrid cycle is optimal. It results in a scalable multigrid preconditioner that can tackle large real-world 3D scenarios.