Sungyun Lee

STABLE FINITE ELEMENT METHODS FOR THE STOKES PROBLEM

The mixed finite element scheme of the Stokes problem with pressure stabi- lization is analyzed for the cross-grid Pk −Pk−1 elements, k ≥ 1, using discontinuous pres- sures. The P + k −Pk−1 elements are also analyzed. We prove the stability of the scheme using the macroelement technique. The order of convergence follows from the standard theory of mixed methods. The macroelement technique can also be applicable to the stability analysis for some higher order methods using continuous pressures such as Taylor-Hood methods, cross-grid methods, or iso-grid methods.

Stable nonconforming quadrilateral finite elements for the Stokes problem

Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Douglas et al. [J. Douglas Jr., J.E. Santos, D.S. Xiu Ye, Nonconforming Galerkin methods based on Quadrilateral Elements for second Order Elliptic Problems, Preprints] added by conforming bubbles to the velocity and discontinuous piecewise linear to the pressure on quadrilateral elements. Optimal order error estimates derived.

Modified mini finite element for the stokes problem in R-2 or R-3

We analyze a modified version of the Mini finite element (or the Mini* finite element) for the Stokes problem in ℝ2 or ℝ3. The cross‐grid element of order one in ℝ3 is also analyzed. The stability is verified with the aid of the macroelement technique introduced by Stenberg. Each of these methods converges with first order in h as the Mini element does. Numerical tests are given for the Mini* element in comparison with the Mini element when Ω is a unit square on ℝ2.

SUPERCONVERGENCE OF FINITE ELEMENT METHOD FOR PARABOLIC PROBLEM

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

Multigrid algorithm for cell centered finite difference on triangular meshes1This work was partially supported by KOSEF, contract number 971-0106-342-2. 1

We consider a multigrid algorithm for the cell centered difference scheme on triangular meshes using a new prolongation operator. The energy norm of this prolongation is shown to be less than 2. Thus the W-cycle is guaranteed to converge. Numerical experiments show that our operator is better than the trivial injection.

Least-squares mixed method for second-order elliptic problems

A theoretical analysis of a least-squares mixed finite element method for second-order elliptic problems having non-symmetric matrix of coefficients is presented. It is proved that the method is not subject to the Ladyzhenskaya-Babuska-Brezzi (LBB) condition and that the finite element approximation yields a symmetric positive definite linear system with condition number O(h^-^2). Optimal error estimates are developed.

A parallel iterative Galerkin method based on nonconforming quadrilateral elements for second-order partial differential equations

A parallel iterative Galerkin method based on domain decomposition technique with nonconforming quadrilateral finite elements will be analyzed for second-order elliptic equations subject to the Robin boundary condition. Optimal order error estimates are derived with respect to a broken H^1-norm and L^2-norm. Applications to time-dependent problems will be considered. Some numerical experiments supporting the theoretical results will be given. This paper is to extend the work in [J. Douglas Jr., J.E. Santos, D. Sheen, X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems, Mathematical Modelling and Numerical Analysis, RAIRO, Model. Math. Anal. Numer. 33 (4) (1999) 747] to the non-self-adjoint case of second-order equations including the term b.@?u. We suppose that uniformly ellipticity holds. Hence the arguments in (loc. cit.) may be applied, word for word. So some proofs will be omitted.

SUPERCONVERGENCE OF A FINITE ELEMENT METHOD FOR LINEAR INTEGRO-DIFFERENTIAL PROBLEMS

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. For k> 1, we obtain first order gain in Lp(2 ≤ p ≤∞ ) norm, second order in W 1,p (2 ≤ p ≤∞ ) norm and almost second order in W 1,∞ norm. For k = 1, we obtain first order gain in W 1,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 the Lp and W 1,p (2 ≤ p ≤∞ ).

Three-dimensional stable nonconforming parallelepiped elements for the Stokes problem

Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Jim Douglas Jr. et al. [Math. Model. Numer. Anal. 33 (4) (1999) 747] for the velocity with conforming bubble functions and discontinuous piecewise linear for the pressure on parallelepiped elements. Optimal order H^1 and L^2 error estimates are derived.