Galerkin Finite Element Methods for Parabolic Problems

² ds accurate of order Anal analysis apply approximate solution assume backward Euler method boundary conditions boundary value problem Ch² Chapter completes the proof consider corresponding data error estimate defined denote differential equations discretization scheme elliptic operator En(t error bound exact solution 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 Math matrix maximum-norm methods for parabolic nonsmooth data error Numer obtain optimal order order q parabolic equations parabolic problem piecewise linear polynomials positive definite proof of Theorem rational function recall result satisfies selfadjoint semidiscrete problem semigroup semilinear SIAM solution operator stability superconvergence Thomée triangle u(tn Uh,t un(t Vx E Sh Wahlbin XESh