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Galerkin-finite element solution of nonlinear evolution problems. (English) Zbl 0549.65079
Differential equations and their applications, Equadiff 5, Proc. 5th Czech. Conf., Bratislava 1981, Teubner-Texte Math. 47, 378-386 (1982).
[For the entire collection see Zbl 0507.00006.]
The generalized Galerkin method for the solution of evolution problems consists of the following steps 1) Formulation in a variational form; 2) Discretization in space; 3) Solving the resulting system of ordinary differential equations. In case of nonlinear problems the application of linear multistep methods has advantage in that we are often able to linearize the resulting scheme without lowering the accuracy. We restrict ourselves to a narrow class of linear multistep methods: to \(A\)-stable methods. These methods lead to unconditionally stable schemes fulfilling certain energy inequalities. Both these properties are desirable, the other providing a simple way for the derivation of apriori error estimates.
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65L20 Stability and convergence of numerical methods for ordinary differential equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations