Search

Two conditions are needed to solve numerical weather prediction models: initial condition and boundary condition. The initial condition has an especially important bearing on the model performance. To get a good initial condition, many data assimilation techniques have been developed for the meteorological and the oceanographical fields. Currently, the most commonly used technique for operational applications is the 3 dimensional (3-D) or 4 dimensional variational data assimilation method. The numerical method used for the cost function minimising process is usually an iterative method such as the conjugate gradient. In this paper, we use the multigrid method based on the cell-centred finite difference on the variational data assimilation to improve the performance of the minimisation procedure for 3D-Var data assimilation. Keywords : numerical weather prediction, variational data assimilation, minimization procedure, multigrid methods, cell centered finite difference (Published: 8 July 2014) Citation : Tellus A 2014, 66 , 20217, http://dx.doi.org/10.3402/tellusa.v66.20217

We consider the approximation of eigenvalue problems for elasticity equations with interface. This kind of problems can be efficiently discretized by using immersed finite element method (IFEM) based on Crouzeix-Raviart P1-nonconforming element. The stability and the optimal convergence of IFEM for solving eigenvalue problems with interface are proved by adapting spectral analysis methods for the classical eigenvalue problem. Numerical experiments demonstrate our theoretical results., 17 pages, 11 figures, 1 table

A GENERALIZATION OF MULTIPLIER THEOREM TO THE BALL

We consider the approximation of elliptic eigenvalue problem with an interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix–Raviart P 1 -nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results.

We study superconvergence of a semi-discrete finite element scheme for parabolic problem. Our new scheme is based on introducing different approximation of initial condition. First, we give a superconvergence ofuh−Rhu, then use a postprocessing to improve the accuracy to higher order.

We introduce a new way of approximating initial condition to the semidiscrete finite element method for integro-differential equations using any degree of elements. We obtain several superconvergence results for the error between the approximate solution and the Ritz-Volterra projection of the exact solution. Fork>1, we obtain first order gain inLp(2≤p≤∞)norm, second order inW1,p(2≤p≤∞)norm and almost second order inW1,∞norm. Fork=1, we obtain first order gain inW1,p(2≤p≤∞)norms. Further, applying interpolated postprocessing technique to the approximate solution, we get one order global superconvergence between the exact solution and the interpolation of the approximate solution in theLpandW1,p(2≤p≤∞).

Systems of nonlinear algebraic equations with a parameter arises in many branches of mathematics and engineering. Computation of the solution curve is usually hindered by the existence of a bifurcation point, where the rank of the corresponding matrix is not full and usually there is another branch curve passing it. We introduce a method from algebraic geometry to find the branch curve. A computational example is provided.

To estimate convergence of the multigrid algorithms, we need some assumptions on smoothers. The assumptions for typical smoothers are well analyzed in the multigrid literature [1,2]. However, numerical evidence shows that Kaczmarz smoother does not satisfy above assumptions. Thus, we introduce a weaker condition which is satisfied by Kaczmarz smoother as well as Jacobi and Gauss-Seidel smoother. Under these weaker assumptions, we show that the convergence factor of V-cycle multigrid algorithm is δ = 1 − 1 (C(j − 1)) . assumptions for Kaczmarz smoother are verified by numerical experiment.