Discrete least-squares finite element methods

Abstract

A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number $\mathcal{O}(h^{-1})$ for a variety of formulations of Poisson's equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson's equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson's equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion.

30 pages

References in corpus (2)

• An ultraweak DPG method for viscoelastic fluids
• Using a DPG method to validate DMA experimental calibration of viscoelastic materials

Cited by in corpus (13)

• High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations
• The Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement for Compressible Viscous Flows
• A Least-Squares Formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement
• Isogeometric Residual Minimization Method (iGRM) with Direction Splitting for Non-Stationary Advection-Diffusion Problems
• On perfectly matched layers for discontinuous Petrov-Galerkin methods
• The DPG-star method
• Goal-oriented adaptive mesh refinement for non-symmetric functional settings
• Using a DPG method to validate DMA experimental calibration of viscoelastic materials
• A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations
• Goal-Oriented Error Estimation for the Automatic Variationally Stable FE Method for Convection-Dominated Diffusion Problems
• Optimal approximation spaces for discontinuous Petrov-Galerkin finite element methods
• An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov equation
• Automatic Variationally Stable Analysis for Finite Element Computations: Transient Convection-Diffusion Problems