Galerkin Finite Element Methods for Parabolic Problems

accurate of order apply approximate solution assume backward Euler method boundary conditions Ch² Chapter completes the proof consider corresponding Crank-Nicolson method data error estimate defined denote differential equation discretization scheme elliptic operator En(t error bound exact solution f(un finite element finite element method follows Galerkin method Green's formula heat equation hence Hilbert space homogeneous equation inequality initial data inner product integration iterative L2-norm Lemma matrix maximum-norm nonsmooth data error obtain optimal order order q parabolic equation parabolic problem piecewise linear polynomials positive definite proof of Theorem rational function recall result satisfies selfadjoint semidiscrete problem semigroup smooth solution operator stability superconvergent term Theorem Thomée triangle u(tn uh,t un(t value problem vanish XESh θε ΘΩ λο ση