Explore the vast range of titles available.

We present the results of an error analysis of a B-spline based finite-element approximation of the stream-function formulation of the large scale wind-driven ocean circulation. In particular, we derive optimal error estimates for h -refinement using a Nitsche-type variational formulations of the two simplied linear models of the stationary quasigeostrophic equations, namely the Stommel and Stommel–Munk models. Numerical results obtained from simulations performed on rectangular and embedded geometries confirm the error analysis.

The embedded discontinuous Galerkin (EDG) method by Cockburn, Gopalakrishnan and Lazarov [B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second-order elliptic problems, SIAM J. Numer. Anal. 47 2009, 2, 1319–1365] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second-order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees , , ( ) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.

This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only can these IFE methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the H1-norm and the L2-norm do not deteriorate when the mesh becomes finer, which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods but are also of a great potential to be useful in error analysis for other related IFE methods.

This paper makes several important contributions to the literature about non- parametric instrumental variables (NPIV ) estimation and inference on a structural function h0 and functionals of h0 .First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series two-stage least squares) estimators of h0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating h0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal root-mean-squared rates for severely ill-posed problems, and are only a logarithmic factor slower than the optimal root-mean- squared rates for mildly ill-posed problems. Third, we use our sup-norm rates to establish the uniform Gaussian process strong approximations and the score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of h0 under primitive conditions, allowing for mildly and severely ill-posed problems. Fourth, as applications, we obtain the first asymptotic pointwise and uniform inference results for plug-in sieve t -statistics of exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low-level conditions when demand is estimated via sieve NPIV. Our real data application of UCBs for exact CS and DL functionals of gasoline demand reveals interesting patterns and is applicable to other goods markets.

By constructing an objective function and using the gradient search, a gradient-based iteration is established for solving the coupled matrix equations AiXBi = Fi, i = 1, 2, …, p. The authors prove that the gradient solution is convergent for any initial values. By analysing the spectral radius of the iterative matrix, the authors obtain an optimal convergence factor. An example is provided to illustrate the effectiveness of the proposed algorithm and to testify the conclusions established in this study.

We propose a discontinuous Galerkin method for fractional convection-diffusion equations with a superdiffusion operator of order α(1 < α < 2) defined through the fractional Laplacian. The fractional operator of order α is expressed as a composite of first order derivatives and a fractional integral of order 2 – α. The fractional convection-diffusion problem is expressed as a system of low order differential/integral equations, and a local discontinuous Galerkin method scheme is proposed for the equations. We prove stability and optimal order of convergence 𝓞(hk+1) for the fractional diffusion problem, and an order of convergence of 𝓞(hk+½) is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.

We prove stability bounds for Stokes-like virtual element spaces in two and three dimensions. Such bounds are also instrumental in deriving optimal interpolation estimates. Furthermore, we develop some numerical tests in order to investigate the behaviour of the stability constants also from the practical side.

A stabilized element-free Galerkin (EFG) method is developed and analyzed in this paper for the meshless numerical solution of Stokes problems. To accelerate the solution procedure and recover the optimal convergence impaired by Gauss integration, integration constraints of Galerkin numerical methods for Stokes problems are derived, and then the inherently consistent reproducing kernel gradient smoothing integration is incorporated into the EFG method with explicit quadrature rules in the reference space. By using Nitsche’s method to satisfy the Dirichlet boundary condition, the inf-sup stability, the existence and uniqueness, and the error estimation of the EFG solution with numerical integration are derived rigorously. Theoretical results reveal that the EFG error essentially comes from not only the meshless approximations of velocity and pressure, but also numerical integration of the Galerkin weak forms. It turns out a procedure on how to choose quadrature rules to ensure that the optimal convergence is not affected by the integration error. Numerical results demonstrate the consistency, efficiency and optimal convergence of the method, and support the theoretical results.

In this paper we present a second order accurate, energy stable numerical scheme for the epitaxial thin film model without slope selection, with a mixed finite element approximation in space. In particular, an explicit treatment of the nonlinear term, ∇ u 1 + | ∇ u | 2 , greatly simplifies the computational effort; only one linear equation with constant coefficients needs to be solved at each time step. Meanwhile, a second order Douglas–Dupont regularization term, A τ Δ 2 ( u n + 1 - u n ) , is added in the numerical scheme, so that an unconditional long time energy stability is assured. In turn, we perform an ℓ ∞ ( 0 , T ; L 2 ) convergence analysis for the proposed scheme, with an O ( τ 2 + h q ) error estimate derived. In addition, an optimal convergence analysis is provided for the nonlinear term using Q q finite elements, which shows that the spatial convergence order can be improved to q + 1 on regular rectangular mesh. A few numerical experiments are presented, which confirms the efficiency and accuracy of the proposed second order numerical scheme.