Max-Norm Estimates for Galerkin Approximations of One-Dimensional Elliptic,Parabolic and Hyperbolic Problems with Mixed Boundary Conditions

The Galerkin methods are studied for two-point boundary value problems and the related one-dimensional parabolic and hyperbolic problems. The boundary value problem considered here is of non-adjoint from and with mixed boundary conditions. The optimal order error estimate in the max-norm is first derived for the boundary problem for the finite element subspace. This result then gives optimal order max-norm error estimates for the continuous and discrete time approximations for the evolution problems described above.