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Resolvent estimates for elliptic finite element operators in one dimension. (English) Zbl 0806.65096
The paper deals with an elliptic one-dimensional operator \(Au= -(au')'+ bu'+ c\), where the coefficients \(a\), \(b\), \(c\) are smooth functions defined on \(\Omega= (0,1)\) and the Dirichlet boundary condition \(u(0)= u(1)= 0\) is assumed. Let \(S_ h\subset H^ 1_ 0(\Omega)\) be the space of spline functions connected with a decomposition of \(\Omega\) with diameter \(h\) and let \(A_ h\) be the corresponding finite element approximation of \(A\). In the first part of the paper some \(L_ p\)- estimates \((0\leq p\leq \infty)\) for the operators \((\lambda I+ A)^{-1}\) and \((\lambda I+ A_ h)^{-1}\), with complex \(\lambda\), are proved.
The second part is devoted to the approximate solution of the parabolic problem \(u_ t+ Au= f(x,t)\), \(u(0,t)= u(1,t)= 0\), \(u(x,0)= u_ 0(x)\) for \(x\in \Omega\), \(t>0\). Error estimates are given for the following approximate problems: (i) semidiscretization by the finite element method with respect to \(x\) and with continuous time, (ii) full discretization by the finite element method with respect to \(x\) and by various methods, particularly by the Euler backward difference method and by the Crank- Nicolson scheme, with respect to \(t\).

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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