A lumped mass finite-element method with quadrature for a nonlinear parabolic problem.

*(English)*Zbl 0591.65079The 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

##### MSC:

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 |