Numerische Mathematik最新文献

The Scott–Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order k and a discontinuous pressure approximation of order (k-1). It employs a “singular distance” (measured by some geometric mesh quantity ( Theta left( textbf{z}right) ge 0) for triangle vertices (textbf{z})) and imposes a local side condition on the pressure space associated to vertices (textbf{z}) with (Theta left( textbf{z}right) =0). The method is inf-sup stable for any fixed regular triangulation and (kge 4). However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices (0<Theta left( textbf{z}right) ll 1). In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the “singular distance”. We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence.

This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (i.e., non-constant density and viscosity) incompressible Navier–Stokes system on a bounded domain. The proposed method is classical in the sense that it consists of a version of the Lax–Friedrichs explicit scheme for the transport equation and a version of Ladyzhenskaya’s implicit scheme for the Navier–Stokes equations. Under the condition that the initial density profile is strictly away from zero, the scheme is proven to be strongly convergent to a weak solution (up to a subsequence) within an arbitrary time interval, which can be seen as a proof of existence of a weak solution to the system. The results contain a new Aubin–Lions–Simon type compactness method with an interpolation inequality between strong norms of the velocity and a weak norm of the product of the density and velocity.

We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

In this paper we establish rigorously a one dimensional model of a junction of several ferromagnetic nanowires. Such structures appear in nano electronics or in new memory devices. We present also a numerical scheme adapted to this configuration and we compare our results with the 3d simulation obtained with the code EMicroM.

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension d. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber k with a number of degrees of freedom scaling as (k^{d-1/2}), which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as (k^d) to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation.

We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is (O(varepsilon )) with (0 < varepsilon ll 1) a dimensionless parameter up to the time at (O(1/varepsilon ^2)). The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discretization at (O(varepsilon ^2tau )) instead of (O(tau )) according to classical error estimates and at (O(h^m+varepsilon ^2tau )) for the full-discretization up to the time (T_{varepsilon } = T/varepsilon ^2) with (T>0) fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the nonlinear Klein–Gordon equation with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with (O(varepsilon ^2)) wavelength in time and (O(varepsilon ^{-2})) wave speed, which indicates that the temporal error is independent of (varepsilon ) when the time step size is chosen as (O(varepsilon ^2)). Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp.

We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace’s equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in (mathbb {R}^d), (dge 2), in the space (L^2(Gamma )), where (Gamma ) denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace’s equation (involving the double-layer potential and its adjoint) cannot be written as the sum of a coercive operator and a compact operator in the space (L^2(Gamma )). Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in ({L^2(Gamma )}) do not converge when applied to the standard second-kind formulations, but do converge for the new formulations.

In this paper we devise and analyze a pressure-robust and superconvergent HDG method in stress-velocity formulation for the Stokes equations and the Navier–Stokes equations with strongly symmetric stress. The stress and velocity are approximated using piecewise polynomial space of order k and (H(textrm{div};Omega ))-conforming space of order (k+1), respectively, where k is the polynomial order. In contrast, the tangential trace of the velocity is approximated using piecewise polynomials of order k. Moreover, the characterization of the proposed schemes shows that the globally coupled unknowns are the normal trace and the tangential trace of velocity, and the piecewise constant approximation for the trace of the stress. The discrete (H^1)-stability is established for the discrete solution. The proposed formulation yields divergence-free velocity, but causes difficulties for the derivation of the pressure-independent error estimate given that the pressure variable is not employed explicitly in the discrete formulation. This difficulty can be overcome by observing that the (L^2) projection to the stress space has a nice commuting property. Moreover, superconvergence for velocity in discrete (H^1)-norm is obtained, with regard to the degrees of freedom of the globally coupled unknowns. Then the convergence of the discrete solution to the weak solution for the Navier–Stokes equations via the compactness argument is rigorously analyzed under minimal regularity assumption. The strong convergence for velocity and stress is proved. Importantly, the strong convergence for velocity in discrete (H^1)-norm is achieved. Several numerical experiments are carried out to confirm the proposed theories.