We consider the effect of numerical integration in finite element  procedures applied to a nonlinear system of two coupled partial differential equations describing the miscible displacement of one incompressible fluid by another in a porous meduim. We consider the use of the numerical quadrature scheme for approximating the pressure and velocity by a mixed method using Raviart - Thomas space of index  and the concentration by a standard Galerkin method. We also give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order estimates are derived when the imposed external flows are smoothly distributed.

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The effect of numerical integration in mixed finite element approximation in the simulation of miscible displacement

Fully discrete IPDG-HMM for multiscale Richards equation of unsaturated flow in porous media

We consider the effect of numerical integration in finite element methods for general nonlinear hyperbolic equations and give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order L2-error estimates are derived.

The effect of numerical integration in finite element methods for general nonlinear hyperbolic equations

A linear triangular finite element method-implicit difference scheme (FEM-IDS) for the solution of second order strongly damped wave equation (SDWE) with memory on domain with interfaces is proposed. Sufficient conditions that guarantee the existence of a unique solution are given. The finite element discretization is such that the arbitrary (but smooth) interface is first approximated by a polygon with a boundary whose vertices all lie on the interface. With the interface being at σ -distance from the approximate interface, linear interpolation operator incorporating the influence of σ isproved. This together with Ritz–Volterra operator and some auxiliary error estimates in the neighborhood of the interface are used to obtain the convergence estimate of the proposed scheme. The scheme is applied to some test examples and the numerical results confirm the computational efficiency of the method.

FEM-IDS for a second order strongly damped wave equation with memory

The effect of numerical integration in finite element methods applied to a class of non-linear parabolic equations is considered and some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration are given. Optimal L2 and H1 estimates for the error and its time derivative are established.

N'guimbi Germain

Papers

The effect of numerical integration in finite element methods for nonlinear parabolic equations