Calcolo最新文献

In this work, three discontinuous Galerkin (DG) methods are formulated and analysed to solve Stokes equations with power law slip boundary condition. Numerical examples exhibited confirm the theoretical findings, moreover we also test the methods on the lid Driven cavity and compare the three DG methods.

Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples m. Our work focuses on providing theoretical approximation guarantees for the class of so-called ((varvec{b},varepsilon ))-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of m-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.

In an open, bounded Lipschitz polygon (Omega subset mathbb {R}^2), we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in (Omega ). The boundary conditions on each edge of (partial Omega ) are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in (H^1(Omega )), quantized tensor formats and certain deep neural networks.

We present a Hermite-type virtual element method with interior penalty to solve the fourth-order elliptic problem over general polygonal meshes, where some interior penalty terms are added to impose the (C^1) continuity. A (C^0)-continuous Hermite-type virtual element with local (H^2) regularity is constructed, such that it can be used in the interior penalty scheme. We prove the boundedness of basis functions and interpolation error estimates of Hermite-type virtual element. After introducing a discrete energy norm, we present the optimal convergence of the interior penalty scheme. Compared with some existing methods, the proposed interior penalty method uses fewer degrees of freedom. Finally, we verify the theoretical results through some numerical examples.

A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are (O(tau ^2 + h^2)) for the velocity in the (L^infty )-norm and the surface elevation in the (L^2)-norm, where (tau ) denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values.

Abstract

In this paper, we first define the weakly essentially irreducible nonnegative tensors and unify the definitions of essentially positive tensors, weakly positive tensors and generalized weakly positive tensors. Then an algorithm to find the spectral radius of weakly essentially irreducible nonnegative tensors is given based on the implicit translational transformation, and the linear convergence condition of the algorithm is analyzed using the directed graph of the matrix, and finally the computational efficiency of the related algorithms is compared using numerical examples.

In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in (L^2(I;L^2(Omega )^d)) and (L^2(I;H^1(Omega )^d)) norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type (L^2(I;H^1(Omega ))) error estimate are new even for scalar parabolic problems.

Abstract

A convection–diffusion problem with a large shift in space is considered. Numerical analysis of high order finite element methods on layer-adapted Durán type meshes, as well as on coarser Durán type meshes in places where weak layers appear, is provided. The theoretical results are confirmed by numerical experiments.

The aim of this paper is twofold. First, we prove (L^p) estimates for a regularized Green’s function in three dimensions. We then establish new estimates for the discrete Green’s function and obtain some positivity results. In particular, we prove that the discrete Green’s functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter h. Actually, we show that at the singularity the discrete Green’s function is of order (h^{-1}), which is consistent with the behavior of the continuous Green’s function. In addition, we also show that the discrete Green’s function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green’s function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.

In this manuscript, a new class of high-order multistep methods on the basis of hybrid backward differentiation formulas (BDF) have been illustrated for the numerical solutions of systems of ordinary differential equations (ODEs) arising from semi-discretization of time dependent partial differential equations. Order and stability analysis of the methods have been discussed in detail. By using an off-step point together with a step point in the first derivative of the solution, the new methods obtained are A-stable for order p, ((p=4,5,6,7)) and (A(alpha ))-stable for order p, ((p=8,9,ldots , 14).) Compared to the existing BDF based method, i.e. class (2+1,) hybrid BDF methods (HBDF), super-future points based methods (SFPBM) and MEBDF, there is a good improvement regarding to absolute stability regions and orders. Some numerical examples are given in order to check the advantage of these methods in reducing the CPU time and thus in increasing accuracy of low and high order the new methods compared to those of SFPBM and MEBDF.