Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations.
Let us put \((Gy)(t) = f(t,y(t))+\int^ t_ 0 K(k,s,y(s))ds\). In this paper, numerical solutions of the nonlinear Volterra initial-value problem (1) \(y'(t) = (Gy)(t)\) in \(I = ]0,T[\), \(y(0) = 0\) are studied under the assumption that functions \(f,k\) are Lipschitz with respect to the variable \(y\). Consider nodes \(0=t_0 < t_1 < \dots < t_N = T\) and intervals \(\sigma_k =]t_k, t_{k+1}[\), \(k=0,1,\dots ,N-1\). For any \(m=1,2,\dots\) and for \(V = \{v\in H^ 1(I); \,v(0)=0\}\), put \(S_m = \{v\in V;\, v/_{\sigma _k} \in P_m\), \(0\leq k < N\}\) and \(DS_{m-1} = \{v\in L_2(I); \, v/_{\sigma _k} \in P_{m-1}\), \(0\leq k < N\}\). A Petrov-Galerkin finite element (PGFE) approximation of the exact solution \(y\) of (1) is a function \(u\in S_m\) satisfying \((u',v) = (Gu, v)\) for all \(v\in DS_{m-1}\). The function \(u_{it}(t) = \int^ t_ 0 (Gu) (s)ds\) is called an iterated PGFE solution of (1). The authors describe several corrections of the approximate solutions \(u\) and \(u_{it}\). They prove estimates of approximation errors of \(y\) and \(y'\) by means of these corrections and present examples illustrating that orders of the error estimates are optimal.
scientific article; zbMATH DE number 1516993
scientific article