A class of mixed finite element methods based on the Helmholtz decomposition in computational mechanics

Abstract

The construction of non-conforming nite element methods (FEMs) for the numerical treatment of partial di erential equations arising in mechanics dates back to the 1960s and is motivated by robust discretizations for almost incompressible materials in solid mechanics, by pointwise divergence-free ansatz functions in uid mechanics, and by low-order ansatz spaces for higher-order problems as the biharmonic problem for the Kirchho plate in structural mechanics. A natural generalization to higher polynomial degrees which preserves the inherent properties of the discretizations is not known so far. This thesis generalizes the non-conforming FEMs of Morley and Crouzeix and Raviart by novel mixed formulations for the Poisson problem, the Stokes equations, the Navier-Lame equations of linear elasticity, and mth-Laplace equations of the form (−1)m∆mu = f for arbitrary m = 1, 2, 3, . . . These formulations are based on Helmholtz decompositions which decompose an unstructured vector eld into a gradient and a curl. The new formulations allow for ansatz spaces of arbitrary polynomial degree and its discretizations coincide with the mentioned non-conforming FEMs for the lowest polynomial degree. Also for higher polynomial degrees, this results in robust discretizations for almost incompressible materials and approximations of the solution of the Stokes equations with pointwise mass conservation. Furthermore this approach also allows for a generalization of the non-conforming FEMs for the Poisson problem and the biharmonic equation to mth-Laplace equations for arbitrary m ≥ 3. A new Helmholtz decomposition for tensor-valued functions enables this. The discretizations presented in this thesis allow not only for a uniform implementation for arbitrary m, but they also allow for lowest-order ansatz spaces, e.g., piecewise a ne polynomials for arbitrary m. The presence of singularities usually a ects the convergence such that higher polynomial degrees in the ansatz spaces show the same convergence rate on uniform meshes as lower polynomial degrees. Therefore adaptive mesh-generation is indispensable especially for ansatz spaces of higher polynomial degree. Besides the a priori and a posteriori analysis, this thesis proves optimal convergence rates for adaptive algorithms for the new discretizations of the Poisson problem, the Stokes equations, and mth-Laplace equations. This is also demonstrated in the numerical experiments of this thesis.