A holomorphic semigroup approach to the lumped mass finite element method
The lumped mass finite element discretization of parabolic equations replaces the functions in the variational form with the time derivative by its lumped counterparts. Optimal error estimates in \(L^2(\Omega)\) and \(H^1(\Omega)\) using this discretization for linear homogeneous parabolic equations are presented where the second order partial differential operator is strongly elliptic. The domain is assumed to be a convex polyhedron in \(\mathbb{R}^d\), \(d\in\{1,2,3\}\). The continuous-in-time formulation of the equation is considered and linear finite elements are used as discretization in space. Results for finite element discretizations without lumping are generalized. The error estimates are derived using the holomorphic semigroup theory.
scientific article; zbMATH DE number 2091426
scientific article