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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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