Behavior of Lagrange‐Galerkin solutions to the Navier‐Stokes problem for small time increment

We consider two kinds of numerical quadrature formulas of Gauss type and Newton‐Cotes type, which are required in the real computation of Lagrange–Galerkin scheme for the Navier–Stokes problem. The Lagrange–Galerkin scheme with numerical quadrature, which has been used practically but not fully analyzed, is proved to be convergent at least for Gauss type quadrature under a condition on the time increment. As for the scheme with Newton‐Cotes type quadrature, it has more smooth convergent property than that of Gauss type, whose reason is discussed.