Optimal error estimates of a Crank–Nicolson finite element projection method for magnetohydrodynamic equations

A stabilized finite element method for the incompressible magnetohydrodynamic equations

We propose and analyze a stabilized nite element method for the incompressible magnetohydrodynamic equations. The numerical results that we present show a good behavior of our approximation in experiments which are relevant from an industrial viewpoint. We explain in particular in the proof of our convergence theorem why it may be interesting to stabilize the magnetic equation as soon as the hy...

Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations

We consider the initial boundary value problem for the homogeneous time-fractional diffusion equation ∂ t u − ∆u = 0 (0 < α < 1) with initial condition u(x, 0) = v(x) and a homogeneous Dirichlet boundary condition in a bounded polygonal domain Ω. We shall study two semidiscrete approximation schemes, i.e., Galerkin FEM and lumped mass Galerkin FEM, by using piecewise linear functions. We establ...

Error Estimates for a Finite Element Method for the Drift-diffusion Semiconductor Device Equations

In this paper, optimal error estimates are obtained for a method for numerically solving the so-called unipolar model (a one-dimensional simpliied version of the drift-diiusion semiconductor device equations). The numerical method combines a mixed nite element method using a continuous piecewise-linear approximation of the electric eld with an explicit upwinding nite element method using a piec...

Pointwise localized error estimates for parabolic finite element equations

Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations

It is shown that the Ritz projection onto spaces of piecewise linear finite elements is bounded in the Sobolev space, Wp\ for 2 <p < oo. This implies that for functions in W¿ n Wp the error in approximation behaves like 0(h) in Wx, for 2 <p =c oo, and like 0(h2) in Lp, for 2 *íp < oo. In all these cases the additional logarithmic factor previously included in error estimates for linear finite e...