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In an attempt to show maximum-norm stability and smoothing estimates for finite element discretizations of parabolic problems on nonquasi-uniform triangulations 			we consider the lumped mass method with piecewise linear finite elements in one and two space dimensions. By an energy argument we derive resolvent estimate for the 			associated discrete Laplacian, which is then a finite difference operator on an irregular mesh, which show that this generates an analytic semigroup in l_p for p‹∞  			uniformly in the mesh, assuming in the two-dimensional case that the triangulations are of Delaunay type, and with a logarithmic bound for p=∞. 			By a different argument based on a weighted norm estimate for a discrete Green's function this is improved to hold without a logarithmic factor for p=∞ 			in one dimension under a weak mesh-ratio condition. Our estimates are applied to show stability also for time stepping methods.

Spectrally Consistent Approximations to the Matrix Exponent and Their Applications to Boundary Layer Problem