Sur la discrétisation totale des problèmes paraboliques

We set an algorithm for the complete discretization of parabolic problems combining the finite element method for the space variables x and the discontinuous Galerkin method for the time t. By using the Legendre's polynomials in t, we reduce the resolution of the system of qN linear equations, where N is the amount of finite element degrees of freedom and q-1 is the degree of the approximations with respect to t, to two inversions of NxN matrices and one evaluation of a NxN matrix polynomial of degree q by a technic of Horner's type. The systematic character of the coefficients allows an algorithm with q as a parameter.