A priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems
The authors study the parametrized nonlinear equation \(F:\Lambda\times H^1_0(J)\to H^{-1}(J)\) with one parameter \(\lambda\in\Lambda\subset \mathbb{R}\) defined as a nonlinear boundary value problem \[ F(\lambda,u)= 0,\quad (\lambda, u)\in\Lambda\times H^1_0(J),\tag{1} \] \[ (F(\lambda,u),v)= \int_J[a(\lambda, x,u'(x)) v'+ f(\lambda, x,u(x))v]dx,\quad \forall v\in H^1_0(J),\tag{2} \] where \(J:=(b,c)\subset\mathbb{R}\) is a bounded interval, and \(a,b:\Lambda\times J\times\mathbb{R}\to \mathbb{R}\) are sufficiently smooth functions, i.e. \(F\) is a second-order differential operator in divergence form. After redefining (1) and (2) using the Sobolev space \(W^{1, \infty}(J)\), the operator \(F\) becomes as smooth as the functions \(a\) and \(f\), and it is a Fredholm operator in a certain open set whose finite element solutions form smooth manifolds under suitable conditions. Making use of the extended implicit function theorem due to \textit{F. Brezzi}, \textit{J. Rappaz} and \textit{P. A. Raviart} [Numer. Math. 36, 1-25 (1980; Zbl 0488.65021); ibid. 37, 1-28 (1981; Zbl 0525.65036); ibid. 38, 1-30 (1981; Zbl 0525.65037)], the authors develop a thorough theory of a priori error estimates of finite element solutions of (1) on regular branches and on branches around turning points in the case of strongly nonlinear problems, i.e. if \(a(\lambda,x,y)\) in (2) is nonlinear with respect to \(y\). Including strongly nonlinear problems extends essentially the theory of the prior works.
scientific article; zbMATH DE number 1004377