A Priori Error Estimates for Mixed Finite Element θ\theta-Schemes for the Wave Equation

A family of implicit-in-time mixed finite element schemes is presented for the numerical approximation of the acoustic wave equation. The mixed space discretization is based on the displacement form of the wave equation and the time-stepping method employs a three-level one-parameter scheme. A rigorous stability analysis is presented based on energy estimation and sharp stability results are obtained. A convergence analysis is carried out and optimal a priori $L^\infty(L^2)$ error estimates for both displacement and pressure are derived

Properties of the first eigenfunctions of the clamped column equation

We show that the clamped column equation may not possess a positive first eigenfunction. This result discovers the anomalies of some papers in determining the shape of the strongest clamped-clamped column

Two-level compact implicit schemes for three-dimensional parabolic problems.

We derive a class of two-level high-order implicit finite difference schemes for solving three-dimensional parabolic problems with mixed derivatives. The schemes are fourth-order accurate in space and second- or lower-order accurate in time depending on the choice of a weighted average parameter μ. Numerical results with μ=0.5 are presented to confirm the high accuracy of the derived scheme and to compare it with the standard second-order central difference scheme. It is shown that the improvement in accuracy does not come at a higher cost of computation and storage since it is possible to choose the grid parameters so that the present scheme requires less work and memory and gives more accuracy than the standard central difference scheme