This article deals with the approximation of the boundary controls of a 1-D linear wave equation with a variable potential by using a finite difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh-size and the convergence of the approximate controls cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the approximation of the controls. Under certain conditions on the potential, we prove that the convergence of the scheme is ensured if the highest frequencies of the discrete initial data have been previously filtered out. Several filtration procedures are proposed and analyzed. Moreover, we identify a class of (regular) continuous initial data which can be controlled uniformly without any special treatment.

This paper is concerned with the nonconforming finite element discretization of geometric partial differential equations. In specific, we construct a surface Crouzeix-Raviart element on the linear approximated surface, analogous to a flat surface. The optimal error estimations are established even though the presentation of the geometric error. By taking the intrinsic viewpoint of manifolds, we introduce a new superconvergent gradient recovery method for the surface Crouzeix-Raviart element using only the information of discretization surface. The potential of serving as an asymptotically exact {\it a posteriori} error estimator is also exploited. A series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method.

As a consequence of Bloch's theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrodinger operators involves integrals over the Brillouin zone. These integrals are difficult to compute numerically in metals due to discontinuities in the integrand. We perform an error analysis of several widely-used quadrature rules and smearing methods for Brillouin zone integration. We precisely identify the assumptions implicit in these methods and rigorously prove error bounds. Numerical results for two-dimensional periodic systems are also provided. Our results shed light on the properties of these numerical schemes, and provide guidance as to the appropriate choice of numerical parameters.

We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie-Butcher theory of Lie group integrators leads to global error estimates.

We propose a new finite volume scheme for the Euler system of gas dynamics motivated by the model proposed by H. Brenner. Numerical viscosity imposed through upwinding acts on the velocity field rather than on the convected quantities. The resulting numerical method enjoys the crucial properties of the Euler system, in particular positivity of the approximate density and pressure and the minimal entropy principle. In addition, the approximate solutions generate a dissipative measure-valued solutions of the limit system. In particular, the numerical solutions converge to the smooth solution of the system as long as the latter exists.

We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error estimates including the case of non-conforming convex sets are given and optimal convergence rates are shown for the lowest-order case. We provide a posteriori bounds that can be used as error indicators in an adaptive algorithm. Numerical studies are presented.

This article focuses on computing Hamiltonian matrix exponential. Given a Hamiltonian matrix $$\mathcal {H}$$ , it is well-known that the matrix exponential $$e^{\mathcal {H}}$$ is a symplectic matrix and its eigenvalues form reciprocal $$(\lambda ,1/\bar{\lambda })$$ . It is important to take care of the symplectic structure for computing $$e^{\mathcal {H}}$$ . Based on the structure-preserving flow proposed by Kuo et al. (SIAM J Matrix Anal Appl 37:976–1001, 2016), we develop a numerical method for computing the symplectic matrix pair $$(\mathcal {M},\mathcal {L})$$ which represents $$e^{\mathcal {H}}$$ .

This paper investigates the numerical modeling of a time-dependent heat transmission problem by the convolution quadrature boundary element method. It introduces the latest theoretical development into the error analysis of the numerical scheme. Semigroup theory is applied to obtain stability in the spatially semidiscrete scheme. Functional calculus is employed to yield convergence in the fully discrete scheme. We compare the results to a more traditional Laplace domain approach.

We present a framework for the systematic numerical approximation of nonlinear evolution problems with dissipation. The approach is based on rewriting the problem in a canonical form that complies with the underlying energy-dissipation structure. We show that the corresponding weak formulation then allows for a dissipation-preserving approximation by Galerkin methods in space and discontinuous Galerkin methods in time. The proposed methodology is rather general and can be applied to a wide range of applications. This is demonstrated by discussion of some typical examples ranging from diffusive partial differential equations to dissipative Hamiltonian systems.

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the $L^2$-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier--Stokes equations. The novel error estimators only take the $\mathrm{curl}$ of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor--Hood and mini finite element methods.