5 research outputs found
On a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed
For , let be an -uniform linear system. The
transversal number of is the minimum
number of points that intersect every line of . The 2-packing
number of is the maximum number of
lines such that the intersection of any three of them is empty. In [Discrete
Math. 313 (2013), 959--966] Henning and Yeo posed the following question: Is it
true that if is a -uniform linear system then
holds for
all ?. In this paper, some results about of -uniform linear systems
whose 2-packing number is fixed which satisfies the inequality are given
A partially penalty immersed Crouzeix-Raviart finite element method for interface problems
Abstract The elliptic equations with discontinuous coefficients are often used to describe the problems of the multiple materials or fluids with different densities or conductivities or diffusivities. In this paper we develop a partially penalty immersed finite element (PIFE) method on triangular grids for anisotropic flow models, in which the diffusion coefficient is a piecewise definite-positive matrix. The standard linear Crouzeix-Raviart type finite element space is used on non-interface elements and the piecewise linear Crouzeix-Raviart type immersed finite element (IFE) space is constructed on interface elements. The piecewise linear functions satisfying the interface jump conditions are uniquely determined by the integral averages on the edges as degrees of freedom. The PIFE scheme is given based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. The solvability of the method is proved and the optimal error estimates in the energy norm are obtained. Numerical experiments are presented to confirm our theoretical analysis and show that the newly developed PIFE method has optimal-order convergence in the L 2 norm as well. In addition, numerical examples also indicate that this method is valid for both the isotropic and the anisotropic elliptic interface problems