Domain Decomposition for Mixed Finite Elements in Elasticity

Abstract

In this work, we deal with a numerical finite element method to solve linear elasticity equations. To discretize the linear elasticity problem, we have chosen a relatively new mixed formulation, in which we seek displacements from space H(curl) and stresses from space H(divdiv) regarding the infinite-dimensional case. When moving to their discrete approximations, choosing these spaces results in displacements with a continuous tangential component and stresses with a continuous normal-normal component. That's where the name of the method comes from. It is a so-called TDNNS formulation. The TDNNS formulation does not suffer from locking effects and is applicable for nearly incompressible materials and for flat elements.

This work takes the reader through implementing the finite element method to constructing the individual matrices and right-hand sides of the resulting system of linear equations. For the completeness and self-readability of the work, we also provide a listing of the specific basis functions of the lowest polynomial order for triangular and quadrilateral elements in two dimensions and tetrahedrons, prisms and, hexahedrons in three dimensions.

But the main contribution of the work is the domain decomposition applied to hybridized mixed elements in two dimensions. We follow a classic primal domain decomposition method that splits the original global problem into two sets of local independent problems and one global problem defined on the subdomain interface. This global problem requires solving a system with the Schur complement matrix. We continue with preconditioning of this problem, thus constructing an approximation to the Schur complement matrix. Overall, we present two local methods of preconditioning. The first is based on the block structure of the Schur complement matrix by neglecting all off-diagonal blocks. The second is based on the Neumann-Neumann method, approximating the inverse of the Schur complement using local contributions.

To achieve complex methods that lead to an optimal condition number bound of O((1+log(H/h))^2), we are forced to apply also global corrections. To this end, we define two operators. Operator implementing a change of basis mapping from the coarse level to the fine level, and operator of the finite element interpolation on the coarse space. Using these operators gives us two options for implementing a global solver. Combining both, local and global corrections, the Edge-based and BDDC algorithms are presented.

Finally, we conduct numerical experiments in which we compare the properties of the Schur complement matrix for the primal pure displacement method of linear elasticity, which is approximated by continuous functions, and the described mixed method with a continuous tangential component for displacement and a continuous normal-normal component for stresses. For the latter, we apply introduced preconditioners. For the local edge preconditioner, which does not employ any coarse space, the method leads to a condition number bound of order O(1/H). We observe an optimal order O((1+log(H/h))^2) of condition number for methods containing both, the local and coarse problems. Furthermore, such methods are additionally robust for jumps in material coefficients that align with the domain decomposition.