Numerische Mathematik最新文献

Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart ((m=1)) or Morley ((m=2)) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated (L^2) error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability.

We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components ((u_1,textbf{u}_2)=(u,-nabla _textbf{x} u)). The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of (L_2)-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides (L_2)-norms of (nabla _textbf{x} u_1) and (textbf{u}_2), the (graph) norm of U contains the (L_2)-norm of (partial _t u_1 +{{,textrm{div},}}_textbf{x} textbf{u}_2). When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of (textbf{u}_2). In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u. In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of (partial _t u_1 +{{,textrm{div},}}_textbf{x} textbf{u}_2), i.e., of the forcing term (f=(partial _t-Delta _x)u). Numerical results show significantly improved convergence rates.

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method’s basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method’s applicability for challenging high-contrast channeled coefficients.

We consider the Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level, so it is purely data-driven. According to a famous veto, convergence in the worst-case scenario cannot be expected in general. However, by imposing certain conditions on the noisy data, we establish a new convergence result which, in addition, requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we also expand the applied range of the Landweber iteration to cover non-smooth ill-posed inverse problems and to handle the situation that the data is contaminated by various types of noise. Numerical simulations are also reported.

In this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on monotonicity or linearity of the kernel (mu ) or the flux f. We first prove a novel Kuznetsov-type lemma for this class of PDEs and thereby show that the finite volume approximations converge to the entropy solution at the rate of (sqrt{Delta t}) in (L^1(mathbb {R})). To the best of our knowledge, this is the first proof of any type of convergence rate for this class of conservation laws. We also present numerical experiments to illustrate this result.

Given a positive integer d and ({{varvec{a}}}_{1},dots ,{{varvec{a}}}_{r}) a family of vectors in ({{mathbb {R}}}^d), ({k_1{{varvec{a}}}_{1}+dots +k_r{{varvec{a}}}_{r}: k_1,dots ,k_r in {{mathbb {Z}}}}subset {{mathbb {R}}}^d) is the so-called lattice generated by the family. In high dimensional integration, prescribed lattices are used for constructing reliable quadrature schemes. The quadrature points are the lattice points lying on the integration domain, typically the unit hypercube ([0,1)^d) or a rescaled shifted hypercube. It is crucial to have a cost-effective method for enumerating lattice points within such domains. Undeniably, the lack of such fast enumeration procedures hinders the applicability of lattice rules. Existing enumeration procedures exploit intrinsic properties of the lattice at hand, such as periodicity, orthogonality, recurrences, etc. In this paper, we unveil a general-purpose fast lattice enumeration algorithm based on linear programming (named FLE-LP).

This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the (L^infty ) norm from the theory of viscosity solutions which are independent of the regularization parameter (varepsilon ). They allow for the uniform convergence of the solution (u_varepsilon ) to the regularized problem towards the Alexandrov solution u to the Monge–Ampère equation for any nonnegative (L^n) right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the (L^infty ) norm for continuously differentiable finite element approximations of u or (u_varepsilon ).

We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de Rham complexes, and smoother finite element differential forms.

We investigate the use of two-layer networks with the rectified power unit, which is called the (text {ReLU}^k) activation function, for function and derivative approximation. By extending and calibrating the corresponding Barron space, we show that two-layer networks with the (text {ReLU}^k) activation function are well-designed to simultaneously approximate an unknown function and its derivatives. When the measurement is noisy, we propose a Tikhonov type regularization method, and provide error bounds when the regularization parameter is chosen appropriately. Several numerical examples support the efficiency of the proposed approach.