Advanced Search

A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points...

Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of ${𝒲}^{1,\infty }\left({ℒ}^2\right)$ is proved. An ${ℒ}^{\infty }\left({\mathscr{H}}^1\right)$-error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations...

We consider a family of conforming finite element schemes with piecewise polynomial space of degree $k$ in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is $h^k+{\tau }^2$ in the discrete norms of ${ℒ}^{\infty }(0,T;{\mathscr{H}}^1\left(\Omega \right))$ and ${𝒲}^{1,\infty }(0,T;{ℒ}^2\left(\Omega \right))$, where $h$ and $\tau$ are the mesh size of the spatial and temporal discretization, respectively....