Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations

Abstract

Abstract. In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assumed to be quasiuniform. In the present paper we show a resolvent estimate, in one and two space dimensions, under weaker conditions on the triangulations than quasiuniformity. In the two-dimensional case, the bound for the resolvent contains a logarithmic factor.Mathematics Subject Classification. 65M06, 65M12, 65M60.

Discrete maximal parabolic regularity for Galerkin finite element methods

“…For a weighted norm |||·||| = σ N 2 · L 2 (Ω) with the weight σ x0 (x) = |x − x 0 | 2 + h 2 and M h ≤ C|ln h| we established this estimate in [27], and used the corresponding result to obtain interior (local) pointwise estimates. Moreover, the resolvent estimate (49) is known also to hold in L p (Ω) norms on a class of non quasi-uniform meshes as well, see [2].…”

“…In contrast to the continuous case, the limiting cases s, p ∈ {1, ∞} are allowed, which explains the logarithmic factor in (3). We also provide the fully discrete analog of (2) and (3).…”

Maximum-norm stability and maximal $$L^{p}$$ L p regularity of FEMs for parabolic equations with Lipschitz continuous coefficients

“…Along with the approach of analytic semigroup, one may reach more precise analysis of the finite element solution, such as maximum-norm error estimates of semi-discrete Galerkin FEMs [35,36,38], resolvent estimates of elliptic finite element operators [2,3,8], error analysis of fully discrete FEMs for parabolic equations [26,29,35], and the discrete maximal L p regularity [14,15]. A related topic is the space-time maximum-norm stability estimate for inhomogeneous equations (f or g j may not be identically zero): u h L ∞ (Ω×(0,T )) ≤ C T u 0 h L ∞ + C T l h u L ∞ (Ω×(0,T )) , ∀ T > 0.…”

Best approximation property in the $W^1_{\infty }$ norm for finite element methods on graded meshes

“…Finite element L 2 projections are of interest in various contexts, especially in the construction and analysis of finite element methods for parabolic problems [3,4]. Proving stability of L 2 projections in norms such as L ∞ and H 1 has thus far required restrictions on the mesh beyond shape regularity.…”