Adaptive least-squares space-time finite element methods

We consider the numerical solution of an abstract operator equation $Bu=f$ by using a least-squares approach. We assume that $B: X \to Y^*$ is an isomorphism, and that $A : Y \to Y^*$ implies a norm in $Y$, where $X$ and $Y$ are Hilbert spaces. The minimizer of the least-squares functional $\frac{1}{2} \, \| Bu-f \|_{A^{-1}}^2$, i.e., the solution of the operator equation, is then characterized by the gradient equation $Su=B^* A^{-1}f$ with an elliptic and self-adjoint operator $S:=B^* A^{-1} B : X \to X^*$. When introducing the adjoint $p = A^{-1}(f-Bu)$ we end up with a saddle point formulation to be solved numerically by using a mixed finite element method. Based on a discrete inf-sup stability condition we derive related a priori error estimates. While the adjoint $p$ is zero by construction, its approximation $p_h$ serves as a posteriori error indicator to drive an adaptive scheme when discretized appropriately. While this approach can be applied to rather general equations, here we consider second order linear partial differential equations, including the Poisson equation, the heat equation, and the wave equation, in order to demonstrate its potential, which allows to use almost arbitrary space-time finite element methods for the adaptive solution of time-dependent partial differential equations

Parallel Implementation of a Least-Squares Spectral Element Solver for Incomressible Flow Problems

Least-squares spectral element methods are based on two important and successful numerical methods: spectral/{\em hp} element methods and least-squares finite element methods. Least-squares methods lead to symmetric and positive definite algebraic systems which circumvent the Ladyzhenskaya-Babu\v{s}ka-Brezzi stability condition and consequently allow the use of equal order interpolation polynomials for all variables. In this paper, we present results obtained with a parallel implementation of the least-squares spectral element solver on a distributed memory machine (Cray T3E) and on a virtual shared memory machine (SGI Origin 3800)

Time finite element methods for structural dynamics

Time finite element methods are developed for the equations of structural dynamics. The approach employs the time-discontinuous Galerkin method and incorporates stabilizing terms having least-squares form. These enable a general convergence theorem to be proved in a norm stronger than the energy norm. Results are presented from finite difference analyses of the time-discontinuous Galerkin and least-squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics. These results show that, for particular interpolations, the time finite element method exhibits improved accuracy and stability.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/50098/1/1620330206_ftp.pd

Split least-squares finite element methods for linear and nonlinear parabolic problems

AbstractIn this paper, we propose some least-squares finite element procedures for linear and nonlinear parabolic equations based on first-order systems. By selecting the least-squares functional properly each proposed procedure can be split into two independent symmetric positive definite sub-procedures, one of which is for the primary unknown variable u and the other is for the expanded flux unknown variable σ. Optimal order error estimates are developed. Finally we give some numerical examples which are in good agreement with the theoretical analysis

Least-Squares Finite Element Formulation for Fluid-Structure Interaction

Fluid-structure interaction problems prove difficult due to the coupling between fluid and solid behavior. Typically, different theoretical formulations and numerical methods are used to solve fluid and structural problems separately. The least-squares finite element method is capable of accurately solving both fluid and structural problems. This capability allows for a simultaneously coupled fluid structure interaction formulation using a single variational approach to solve complex and nonlinear aeroelasticity problems. The least-squares finite element method was compared to commonly used methods for both structures and fluids individually. The fluid analysis was compared to finite differencing methods and the structural analysis type compared to traditional Weak Galerkin finite element methods. The simultaneous solution method was then applied to aeroelasticity problems with a known solution. Achieving these results required unique iterative methods to balance each domain\u27s or differential equation\u27s weighting factor within the simultaneous solution scheme. The scheme required more computational time but it did provide the first hands-off method capable of solving complex fluid-structure interaction problems using a simultaneous least-squares formulation. A sequential scheme was also examined for coupled problems