In solving the one dimensional heat equation, the solution is approximated using a Galerkin technique. Then, if the approximate solution U is required to lie in a Hermite space of piecewise polynomials of degree 3 based on a rectangular grid of mesh size h ... in either case, it can be shown that the computed solution U satisfies < a constant. The solution is computed using floating point base N-arithmetic with a T digit mantissa and v=N1-T .