The coupled Stokes and Darcy equations are approximated by a strongly conservative finite element method. The discrete spaces are the divergence-conforming velocity space with matching pressure space such as the Raviart-Thomas spaces. This work proves optimal error estimate of the velocity in the L2 norm in the domain and on the interface. Lipschitz regularity of the interface is sufficient to obtain the results.

We propose a duality based a posteriori error estimator for the computation of functionals averaged in time for nonlinear time dependent problems. Such functionals are typically relevant for (quasi-) periodic solutions in time. Applications arise, e. g. in chemical reaction models. In order to reduce the numerical complexity, we use simultaneously locally refined meshes and adaptive (chemical) models. Hence, considerations of adjoint problems measuring the sensitivity of the functional output are needed. In contrast to the classical dual- weighted residual (DWR) method, we favor a fixed mesh and model strategy in time. Taking advantage of the (quasi-) periodic behaviour, only stationary dual problems have to be solved.

Abstract-The main purpose of this paper is to propose the Chebyshev spectral-collocation method for a class of the weakly singular Volterra integral equations (VIEs) with proportional delay. The proposed method also are applicable to a class of the weakly singular VIEs with proportional delay possessing unsmooth solution. To provide a rigorous error analysis for the proposed method, we prove the the uniqueness and smoothness of the solution. The error analysis shows that the numerical errors decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.

In this paper a general parabolic problem is considered and discretized by the discontinuous Galerkin (DG) method in time and, in general, in space. Optimal a priori error estimates in space as well as in time are derived and applied to the heat equation and to a nonlinear convection-diffusion equation.

- Vortex rings are generated in numerous practical applications (biological propulsion, internal combustion engines) by the sudden injection of a fluid. Measurements of such flows often provide partial information, generally limited to the exterior of the vortex ring. We propose in this paper a new algorithm for the reconstruction of the stream function inside a bounded domain, where a vortex ring is supposed to form.We derive a quasi-Newton Broyden- Fletcher-Goldfarb-Shanno (BFGS) algorithm for this problem, using a cost function that describes the matching between the reconstructed field and given values of normal derivatives on the boundary of the domain. The algorithm computes the best parameters of the vortex model that is used to describe the vorticity distribution inside the vortex ring. Several vortex models are considered. The method is implemented in freefem++ using a P1 finite-element spatial discretization. Validation tests using analytical models (Hill’s spherical vortex) and Navier-Stokes direct numerical simulation data show good agreement between reconstructed and reference fields.

Article On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials was published on April 1, 2012 in the journal Journal of Numerical Mathematics (volume 20, issue 1).