Resolvent estimates for elliptic finite element operators in one dimension

Crouzeix, M.; Larsson, S.; Thomée, V.

doi:10.1090/S0025-5718-1994-1242058-1

Resolvent estimates for elliptic finite element operators in one dimension
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by M. Crouzeix, S. Larsson and V. Thomée PDF

Math. Comp. 63 (1994), 121-140 Request permission

Abstract:

We prove the analyticity (uniform in h) of the semigroups generated on ${L_p}(0,1),1 \leq p \leq \infty$, by finite element analogues ${A_h}$ of a one-dimensional second-order elliptic operator A under Dirichlet boundary conditions. This is accomplished by showing the appropriate estimates for the resolvents by means of energy arguments. The results are applied to prove stability and optimal-order error bounds for numerical solutions of the associated parabolic problem for both smooth and nonsmooth data.

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