Discrete least-squares finite element methods

Abstract

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 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. The methodology described here was developed as an outgrowth of the discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [29]. The strength of DPG is highlighted throughout, however, the majority of theory presented is general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

The DPG-star method

“…Notice that the DPG stiffness matrix, B T G −1 B, is always symmetric and positive-definite and that after solving for w via (29), v can always be recovered with only local cost, i.e., v = G −1 (f − Bw). Construction of the stiffness matrix B T G −1 B with broken spaces is considered in detail in [40].…”

“…Consider the general setting of Section 2.3 and the broken ultraweak DPG* formulation which is proved to be well posed in Theorem 2.4. Namely, with L set to the general partial differential operator in (30), the problem of finding a v ∈ H 0 (L) satisfying Lv = f is reformulated as (40), where…”

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

“…We take ν = 0.1 and N x = N y = N in the numerical simulations. The nonuniformly distributed collocation point set S (10,10) Ω is used in the four methods (17), (25), (18), and (26), unless specified otherwise. The choice of d * in the formulations (17), (18), (25), and (26) is as follows.…”

“…The distribution of collocation points is shown as Figure 16 (right), which is defined similarly to S (p,q) Ω ; see (10). where α = β.…”

On perfectly matched layers for discontinuous Petrov–Galerkin methods

“…Our focus will be directed towards discontinuous Petrov-Galerkin (DPG) methods which, by their nature, involve such non-symmetric variational formulations [8]. DPG methods have already been studied for acoustic wave equations in [3,4,6,7,[9][10][11][12][13][14], for Maxwell's equations [15], for elastic media [16][17][18][19][20][21][22][23][24], and even for applications in nonlinear optics with the Schrödinger equation [25]. Until now, the construction of perfectly matched layers (PMLs) for DPG methods have not been significantly analyzed.…”