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A lumped mass finite-element method with quadrature for a nonlinear parabolic problem. (English) Zbl 0591.65079
The authors consider the nonlinear uniformly parabolic initial boundary value problem \[ u_ t-\nabla (a(x,t,u)\nabla u)=f(x,t,u)\quad in\quad \Omega \times [0,T], \] \[ u=0\quad on\quad \partial \Omega \times [0,T],\quad u(.,0)=v\quad in\quad \Omega, \] where \(\Omega\) is a two- dimensional domain with smooth boundary \(\partial \Omega\) and the functions a(x,t,u) and f(x,t,u) are sufficiently smooth. By quasi-uniform triangularization of \(\Omega\) and use of corresponding pyramidal basis functions they formulate the standard semi-discrete Galerkin method in form of a system of ordinary differential equations with time t as independent variable. They simplify this system by approximating inner products by quadrature with equal weights for the vertices of each triangle (separately for each triangle). Denoting by h the maximal diameter of a triangle they show that the approximate solution has error of order \(h^ 2\) in \(L_ 2(\Omega)\)-norm, \(h^ 2 \log (1/h)\) in \(L_{\infty}(\Omega)\)-norm, whereas the gradient has error of order h in the \(L_ 2(\Omega)\)-norm. Then they discretize time t also, using k as time step, and investigate the forward Euler scheme (requiring the usual restriction of time step) and linearized versions of the backward Euler scheme and the Crank-Nicolson scheme, and again errors come out as expected, for the approximate solution in the latter two schemes of orders \(h^ 2+k\) or \(h^ 2+k^ 2\), respectively, in the \(L_ 2(\Omega)\)-norm, with a factor log(1/h) to be adjoined if the \(L_{\infty}\)-norm is used. Finally, monotonicity properties of the discrete solution are discussed.
Reviewer: R.Gorenflo

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N40 Method of lines for boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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