393 research outputs found
On the Singular Neumann Problem in Linear Elasticity
The Neumann problem of linear elasticity is singular with a kernel formed by
the rigid motions of the body. There are several tricks that are commonly used
to obtain a non-singular linear system. However, they often cause reduced
accuracy or lead to poor convergence of the iterative solvers. In this paper,
different well-posed formulations of the problem are studied through
discretization by the finite element method, and preconditioning strategies
based on operator preconditioning are discussed. For each formulation we derive
preconditioners that are independent of the discretization parameter.
Preconditioners that are robust with respect to the first Lam\'e constant are
constructed for the pure displacement formulations, while a preconditioner that
is robust in both Lam\'e constants is constructed for the mixed formulation. It
is shown that, for convergence in the first Sobolev norm, it is crucial to
respect the orthogonality constraint derived from the continuous problem. Based
on this observation a modification to the conjugate gradient method is proposed
that achieves optimal error convergence of the computed solution
Parameter-robust discretization and preconditioning of Biot's consolidation model
Biot's consolidation model in poroelasticity has a number of applications in
science, medicine, and engineering. The model depends on various parameters,
and in practical applications these parameters ranges over several orders of
magnitude. A current challenge is to design discretization techniques and
solution algorithms that are well behaved with respect to these variations. The
purpose of this paper is to study finite element discretizations of this model
and construct block diagonal preconditioners for the discrete Biot systems. The
approach taken here is to consider the stability of the problem in non-standard
or weighted Hilbert spaces and employ the operator preconditioning approach. We
derive preconditioners that are robust with respect to both the variations of
the parameters and the mesh refinement. The parameters of interest are small
time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.Comment: 24 page
Robust preconditioners for PDE-constrained optimization with limited observations
Regularization robust preconditioners for PDE-constrained optimization
problems have been successfully developed. These methods, however, typically
assume that observation data is available throughout the entire domain of the
state equation. For many inverse problems, this is an unrealistic assumption.
In this paper we propose and analyze preconditioners for PDE-constrained
optimization problems with limited observation data, e.g. observations are only
available at the boundary of the solution domain. Our methods are robust with
respect to both the regularization parameter and the mesh size. That is, the
condition number of the preconditioned optimality system is uniformly bounded,
independently of the size of these two parameters. We first consider a
prototypical elliptic control problem and thereafter more general
PDE-constrained optimization problems. Our theoretical findings are illuminated
by several numerical results
Weakly imposed symmetry and robust preconditioners for Biot's consolidation model
We discuss the construction of robust preconditioners for finite element
approximations of Biot's consolidation model in poroelasticity. More precisely,
we study finite element methods based on generalizations of the
Hellinger-Reissner principle of linear elasticity, where the stress tensor is
one of the unknowns. The Biot model has a number of applications in science,
medicine, and engineering. A challenge in many of these applications is that
the model parameters range over several orders of magnitude. Therefore,
discretization procedures which are well behaved with respect to such
variations are needed. The focus of the present paper will be on the
construction of preconditioners, such that the preconditioned discrete systems
are well-conditioned with respect to variations of the model parameters as well
as refinements of the discretization. As a byproduct, we also obtain
preconditioners for linear elasticity that are robust in the incompressible
limit.Comment: 21 page
A mixed finite element method for nearly incompressible multiple-network poroelasticity
In this paper, we present and analyze a new mixed finite element formulation
of a general family of quasi-static multiple-network poroelasticity (MPET)
equations. The MPET equations describe flow and deformation in an elastic
porous medium that is permeated by multiple fluid networks of differing
characteristics. As such, the MPET equations represent a generalization of
Biot's equations, and numerical discretizations of the MPET equations face
similar challenges. Here, we focus on the nearly incompressible case for which
standard mixed finite element discretizations of the MPET equations perform
poorly. Instead, we propose a new mixed finite element formulation based on
introducing an additional total pressure variable. By presenting energy
estimates for the continuous solutions and a priori error estimates for a
family of compatible semi-discretizations, we show that this formulation is
robust in the limits of incompressibility, vanishing storage coefficients, and
vanishing transfer between networks. These theoretical results are corroborated
by numerical experiments. Our primary interest in the MPET equations stems from
the use of these equations in modelling interactions between biological fluids
and tissues in physiological settings. So, we additionally present
physiologically realistic numerical results for blood and tissue fluid flow
interactions in the human brain
Unified Framework for Finite Element Assembly
At the heart of any finite element simulation is the assembly of matrices and
vectors from discrete variational forms. We propose a general interface between
problem-specific and general-purpose components of finite element programs.
This interface is called Unified Form-assembly Code (UFC). A wide range of
finite element problems is covered, including mixed finite elements and
discontinuous Galerkin methods. We discuss how the UFC interface enables
implementations of variational form evaluation to be independent of mesh and
linear algebra components. UFC does not depend on any external libraries, and
is released into the public domain
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