This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main ...comparison result is that the error of the P2P0-FEM is a lower bound to the error of the Bernardi–Raugel (or reduced P2P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs.
Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition.
We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional ...Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.
Summary
Following the so‐called cracking elements method (CEM), we propose a novel Galerkin‐based numerical approach for simulating quasi‐brittle fracture, named global cracking elements method ...(GCEM). For this purpose the formulation of the original CEM is reorganized. The new approach is embedded in the standard framework of the Galerkin‐based finite‐element method (FEM), which uses disconnected element‐wise crack openings for capturing crack initiation and propagation. The similarity between the proposed global cracking elements (GCEs) and the standard nine‐node quadrilateral element (Q9) suggests a special procedure: the degrees of freedom of the center node of the Q9, originally defining the displacements, are “borrowed” to describe the crack openings of the GCE. The proposed approach does not need remeshing, enrichment, or a crack‐tracking strategy, and it avoids a precise description of the crack tip. Several benchmark tests provide evidence that the new approach inherits from the CEM most of the advantages. The numerical stability and robustness of the GCEM are better than the ones of the CEM. However, presently only quadrilateral elements with nonlinear interpolations of the displacement field can be used.
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