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Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations. (English) Zbl 1116.65108
The authors study the finite element semidiscretisation of parabolic equations with one and two spatial dimensions governed by elliptic operators that fulfill a Gårding inequality. The main results are estimates for the resolvent of the discrete elliptic operator in the maximum norm which show that the discrete elliptic operator generates an analytic semigroup and which reflect the parabolic smoothing property. In the two-dimensional case, a logarithmic factor appears. The estimates are proved for regular families of triangulations satisfying assumptions weaker than quasi-uniformity. The proofs are based upon \(L^1\)-estimates of the adjoint discrete Green’s function.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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