Least-squares Finite Element Method Research Articles

MIM: A deep mixed residual method for solving high-order partial differential equations

In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function and a deep neural network (DNN) to approximate the PDE solution. In this work, we propose a deep mixed residual method (MIM) to solve PDEs with high-order derivatives. Notable examples include Poisson equation, Monge-Ampére equation, biharmonic equation, and Korteweg-de Vries equation. In MIM, we first rewrite a high-order PDE into a first-order system, very much in the same spirit as local discontinuous Galerkin method and mixed finite element method in classical numerical methods for PDEs. We then use the residual of the first-order system in the least-squares sense as the loss function, which is in close connection with least-squares finite element method. For aforementioned classical numerical methods, the choice of trial and test functions is important for stability and accuracy issues in many cases. MIM shares this property when DNNs are employed to approximate unknowns functions in the first-order system. In one case, we use nearly the same DNN to approximate all unknown functions and in the other case, we use totally different DNNs for different unknown functions. Numerous results of MIM with different loss functions and different choices of DNNs are given for four types of PDEs. In most cases, MIM provides better approximations (not only for high-order derivatives of the PDE solution but also for the PDE solution itself) than DGM with nearly the same DNN and the same execution time, sometimes by more than one order of magnitude. MIM with multiple DNNs often provides better approximations than MIM with only one DNN, sometimes by more than one order of magnitude. Numerical results also indicate interesting connections between MIM and classical numerical methods. Therefore, we expect MIM to open up a possibly systematic way to understand and improve deep learning for solving PDEs from the perspective of classical numerical analysis.

Adjoint-based optimal control of incompressible flows with convective-like energy-stable open boundary conditions

This paper describes an open-loop optimal control strategy for flow problems governed by the incompressible Navier-Stokes equations with convective-like energy-stable boundary conditions. A quasi-Newton optimization procedure is employed, and the required objective sensitivities are computed using the continuous adjoint method. The adjoint equations for the corresponding system are derived and discussed. Both the primal and adjoint systems are solved using the least-squares finite element method. This choice circumvents the LBB condition and leads to symmetric positive-definite algebraic systems. Numerical examples are provided to show the stability and accuracy of the proposed approach.

Simulating sea ice drift in the Southern Ocean incorporating real wind data using the LSFEM

AbstractThe viscous‐plastic sea ice model of [2] describes the motion of sea ice on large scales. The numerical model of the sea ice drift considers velocities and stresses and is coupled with the field quantities ice thickness and ice concentration, which are modeled by transient advection equations. Here, the viscosity in sense of a non‐Newtonian fluid depends on velocity gradients, as well as on ice concentration and ice thickness. This leads to a strong nonlinearity of the constitutive model.In [8], we discussed the hurdles of large scale sea ice models and possible solutions of an implementation within a least‐squares finite element method (LSFEM). Previous studies on sea ice model implementations have shown that the LSFEM is a promising solution to the numerically difficult problem, cf. [9] and [7]. In this work, another essential aspect of sea ice research comes into play: embedding real data into the numerical solution is an important field of research in sea ice and ocean modeling. Here, we show a 2D boundary value problem for simulating sea ice drift in the Southern Ocean incorporating real wind data from the ERA‐5 project [1].

Least‐squares formulation to solve non‐Newtonian fluid flow and application of data assimilation in 2D

AbstractIn this contribution, we present an approach to model steady flow of incompressible non‐Newtonian fluids with data assimilation into the numerical solution based on the least‐squares finite element method (LSFEM). The assimilation of data (e.g. experimental or analytical) into numerical simulations offers promising possibilities when examining complex problems. Potential applications include the enhancement of numerical models using measured data or the completion of experimental data using numerical methods, e.g. the determination of non‐measured quantities such as pressure. In particular for the field of fluid mechanics, the implemented LSFEM has some theoretical advantages compared to the well‐known (mixed) Galerkin finite element method, since it is not restricted to the LBB‐condition. Additionally, it results in a minimization problem with symmetric positive (semi‐)definite equation systems also for differential equations with non‐self‐adjoint operators. A further advantage in this context is that the assimilation of data can easily be performed by adding a term to the least‐squares (LS) functional such that it does not significantly increase the computational cost. The approach of data assimilation is shown by solving steady flow of a non‐Newtonian fluid through a channel with a smooth contraction.

Improving convergence by optimizing the condition number of the stiffness matrices arising from least-squares finite element methods

The least-squares finite element method (LSFEM) has drawn much attention with desirable properties, such as always symmetric positive-definite stiffness matrix and approximation of primal and non-primal variables with equivalent accuracy. Despite a condition number comparable to that of Galerkin FEM, of O(h−2), it is sometimes found that LSFEM with low-order elements does not show a satisfactory accuracy of solution. In this report, we propose an easy-to-implement technique that optimizes the condition number of the stiffness matrix. The least-squares functional is applied to the first-order governing equations and a linear combination of the weak form domain and boundary equations, with normalization (i.e., weighting) parameters is constructed. The impact of these parameters on the condition number of the resulting global stiffness matrix is proved to be independent of mesh size or shape function order. By use of a global optimization scheme that tunes these parameters, we observe notable decrease in the condition number compared to cases where the least-squares principle is directly applied to the governing equations, and hence convergence is significantly improved. It is further shown that the technique can be applied to the elemental stiffness matrix for a computationally efficient means of determining normalization parameters. To show the general applicability of the method, we also apply this technique to solve for a two-dimensional (2D) lid-driven flow problem and to calculate the electromagnetic field distribution in a 2D rectangular resonant cavity. In both problems, the LSFEM solution accuracy is poor prior to optimization of weighting parameters, and accuracy does not increase much by mesh refinement alone. Moreover, the process can be generalized to Galerkin FEM and other methods to improve convergence properties.

The cubic B-spline least-squares finite-element method for the numerical solutions of regularized long-wave equation

A new least-squares method based on cubic B-spline functions is presented to solve regularized long-wave (RLW) equation numerically. An algebraic system of equations is obtained by employing the least-squares method based on cubic B-spline approximation for spatial discretization and the Crank–Nicolson technique for time discretization for integrating the RLW equation. The two test problems are implemented to show suitability of the proposed scheme. New results have been compared with those of some previous studies available in the literature.

A least-squares finite element method for steady flows across an unconfined square cylinder placed symmetrically in a plane channel

This paper develops a least-squares finite element method (LSFEM) for non-Newtonian flows of power-law fluid across an unconfined square cylinder at two inclination angles, placed symmetrically in a two-dimensional channel. We used Newton's method to linearize the equation of motion and employed the least-squares approach with systematic mesh refinements to flow past the square cylinder. The LSFEM offers a direct approximation of the extra stress tensor components, a symmetric positive definite system, and the openness of choosing finite element spaces. We proved that the least-squares approximation converges to linearized solutions of non-Newtonian fluid flows. We demonstrated that the numerical results using low-order basis functions agree with the theoretical estimates and presented the effects of different physical parameters on the physical attributes of the power-law model at a low Reynolds number, such as drag coefficients, viscosity, and velocity. These results agree with others published in the literature. Hence, the LSFEM in non-Newtonian fluid simulations can accomplish the identification of shear-thinning and shear-thickening characteristics.

Space–time least-squares finite elements for parabolic equations

We present a space–time least-squares finite element method for the heat equation. It is based on residual minimization in L2 norms in space–time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have a local a-posteriori error estimator for free. In particular, our approach features full space–time adaptivity. We also present a-priori error analysis on simplicial space–time meshes which are highly structured. Numerical results conclude this work.

Numerical Investigation of Newtonian Fluid Flow around Sharp & Rounded Corner Rectangular Obstruction

In the present study, very stable and converging Least Square finite element method (LSFEM) is employed to calculate the approximate solution of steady state Navier – Stokes equation, consisting continuity and momentum equations in two dimensions. The 2D rectangular domain is considered to study the behavior of linear fluid passing two parallel rectangular obstructions in an open channel. The current numerical analysis is based on two instances, in the first one two rectangular obstructions possesses sharp corners set to face the entering flow while in the other case, obstructions having round corners are analyzed concerning to examine the enhancement in the size of vortex formed due to the flow blockage, intensity of vortex and the recirculation flow rate in the dead zone for Reynolds number from 250 to 2000. The stream line patterns are also presented to monitor the changes appears in the shape of vortex for different Re. The information gather through the analysis suggests the cutoff corner obstruction is congenial in restricting the vortex length and favorable for slow down the flow circulation rate in the lower stream of the channel. The acquired results are compared with established data in past literature which turn out to be in good agreement.

Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H (div)

Abstract Given an arbitrary function in $\boldsymbol{H}({\operatorname{div}})$, we show that the error attained by the global-best approximation by $\boldsymbol{H}({\operatorname{div}})$-conforming piecewise polynomial Raviart–Thomas–Nédélec elements under additional constraints on the divergence and normal flux on the boundary is, up to a generic constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The generic constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in $\boldsymbol{H}({\operatorname{div}})$, delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size $h$ and the polynomial degree $p$, for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori$hp$-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.

Least-Squares Finite Element Method for a Meso-Scale Model of the Spread of COVID-19

This paper investigates numerical properties of a flux-based finite element method for the discretization of a SEIQRD (susceptible-exposed-infected-quarantined-recovered-deceased) model for the spread of COVID-19. The model is largely based on the SEIRD (susceptible-exposed-infected-recovered-deceased) models developed in recent works, with additional extension by a quarantined compartment of the living population and the resulting first-order system of coupled PDEs is solved by a Least-Squares meso-scale method. We incorporate several data on political measures for the containment of the spread gathered during the course of the year 2020 and develop an indicator that influences the predictions calculated by the method. The numerical experiments conducted show a promising accuracy of predictions of the space-time behavior of the virus compared to the real disease spreading data.

An adaptive least-squares finite element method for Giesekus viscoelastic flow problems

In this study, a least-squares (LS) finite element method with an adaptive mesh approach is investigated for Giesekus viscoelastic flow problems. We consider the weighted LS method on uniform and adaptive meshes for the Newton linearized viscoelastic problem, where adaptive grids are automatically generated by the least-squares solutions. We use a residual-type a-posteriori error estimator to adjust weights in the LS functional and compare the convergence behaviour of adaptive meshes generated using different grading functions. Numerical results demonstrate that the adaptive LS method shows at least the first-order convergence rate when equal-order linear interpolation functions are used for all variables, which agrees with the theoretical estimate. In addition, adaptive grids generated using the velocity outperform those based on the a-posteriori error estimator, yielding better numerical results.

Collective marking for arbitrary order adaptive least-squares finite element methods with optimal rates

The collective marking strategy with alternative refinement-indicators in adaptive mesh-refining of least-squares finite element methods (LSFEMs) has recently been shown to lead to optimal convergence rates in Carstensen (2020). The proofs utilize explicit identities for the lowest-order Raviart–Thomas and the Crouzeix–Raviart finite elements. This paper generalizes those results to arbitrary polynomial degree and mixed boundary conditions with some novel arguments. The analysis is outlined for the Poisson equation in 3D with mixed boundary conditions.

A least‐squares finite element formulation to solve incompressible non‐Newtonian fluid flow

AbstractThe Modeling of non‐Newtonian fluids is an essential issue for many industrial applications, for example in the field of chemistry, bioengineering and medical science. In this contribution we present a least‐squares (LS) finite element approach to model steady flow of incompressible non‐Newtonian fluids based on the Navier‐Stokes equations. The application of the least‐squares finite element method (LSFEM) especially in the case of fluid mechanics is motivated by some theoretical advantages compared to the well‐known (mixed) Galerkin method. The LSFEM is not restricted to the LBB‐condition and results in a minimization problem with symmetric positive definite equation systems also for differential equations with non self‐adjoint operators. Furthermore, the LSFEM provides an inherent a posteriori error estimator without additional costs, which enables an efficiency enhancement through adaptive mesh refinements.In this contribution the first‐order LS formulation in terms of stresses and velocities, as introduced in [4], is extended to consider the nonlinear dependence of the viscosity on the shear rate of the fluid. Therefore, the well‐known Carreau model for generalized Newtonian fluids, see for instance [1], is investigated. For the approximation of the independent variables, velocities and stresses, we apply conforming discretizations in H1 and H(div) using standard Lagrange interpolation polynomials and vector‐valued Raviart‐Thomas interpolations functions. A flow through a square domain with exact solution is considered to validate the proposed schemes with respect to accuracy and efficiency.

Large Scale Sea Ice Modeling – Problems and Perspectives.

AbstractThe visco‐plastic sea ice model based on [7] describes the movement of sea ice over large areas of several thousand square kilometers in time. This model has been considered in many publications and has been extended and adapted by numerically motivated and physically‐based approaches. The basic model for the simulation of sea ice circulation considers sea ice velocities and stresses coupled to the field quantities of sea ice thickness and concentration. Two transient advection equations describe the development of sea ice thickness and concentration coupled with sea ice velocity. Furthermore, the viscosity in the constitutive equation is dependent on the sea ice velocities in the sense of a non‐Newtonian fluid, which makes the constitutive relationship strongly nonlinear. An extension of the model is, for example, the elasto‐visco‐plastic constitutive law proposed by [10], which gives numerical stabilization. Recent research on the finite element implementation of the sea ice model is turned to formulations based on the (mixed) Galerkin variation approach, see for example [1] and [20]. Likewise, in [15], [16], and [18] solvers are proposed for the efficient solution of the problem.In this paper, we discuss the obstacles and possibilities of a sea ice model implementation, among others, within a least‐squares finite element method (LSFEM). The mixed LSFEM is well established in fluid mechanics, and a significant advantage of the method is its applicability to first‐order systems, see e.g. [12]. Thus, this method leads to stable and robust formulations for non‐self‐adjoint systems, as they are, for example, for the tracer equations. Based on the results of the Taylor least‐squares scheme and a first‐order Crank‐Nicolson time integrator scheme for the tracer equations, see [21], we discuss here possible steps towards an adequate solution strategy for the complete sea ice model.

An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows

<p style='text-indent:20px;'>We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns.</p>

Simulation of Ocean Circulation of Dongsha Water Using Non-Hydrostatic Shallow-Water Model

A two-dimensional non-hydrostatic shallow-water model for weakly dispersive waves is developed using the least-squares finite-element method. The model is based on the depth-averaged, nonlinear and non-hydrostatic shallow-water equations. The non-hydrostatic shallow-water equations are solved with the semi-implicit (predictor-corrector) method and least-squares finite-element method. In the predictor step, hydrostatic pressure at the previous step is used as an initial guess and an intermediate velocity field is calculated. In the corrector step, a Poisson equation for the non-hydrostatic pressure is solved and the final velocity and free-surface elevation is corrected for the new time step. The non-hydrostatic shallow-water model is verified and applied to both wave and flow driven fluid flows, including solitary wave propagation in a channel, progressive sinusoidal waves propagation over a submerged bar, von Karmann vortex street, and ocean circulations of Dongsha Atolls. It is found hydrostatic shallow-water model is efficient and accurate for shallow water flows. Non-hydrostatic shallow-water model requires 1.5 to 3.0 more cpu time than hydrostatic shallow-water model for the same simulation. Model simulations reveal that non-hydrostatic pressure gradients could affect the velocity field and free-surface significantly in case where nonlinearity and dispersion are important during the course of wave propagation.

A short note on plain convergence of adaptive least-squares finite element methods

We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract framework which covers a variety of model problems.

Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations

In this paper, two flux-only least-squares finite element methods (LSFEM) for the linear hyperbolic transport problem are developed. The transport equation often has discontinuous solutions and discontinuous inflow boundary conditions, but the normal component of the flux across the mesh interfaces is continuous. In traditional LSFEMs, the continuous finite element space is used to approximate the solution. This will cause unnecessary error around the discontinuity and serious overshooting. In arXiv:1807.01524 [math.NA], we reformulate the equation by introducing a new flux variable to separate the continuity requirements of the flux and the solution. Realizing that the Raviart-Thomas mixed element space has enough degrees of freedom to approximate both the flux and its divergence, we eliminate the solution from the system and get two flux-only formulations, and develop corresponding LSFEMs. The solution then is recovered by simple post-processing methods using its relation with the flux. These two versions of flux-only LSFEMs use less DOFs than the method we developed in arXiv:1807.01524 [math.NA]. Similar to the LSFEM developed in arXiv:1807.01524 [math.NA], both flux-only LSFEMs can handle discontinuous solutions better than the traditional continuous polynomial approximations. We show the existence, uniqueness, a priori and a posteriori error estimates of the proposed methods. With adaptive mesh refinements driven by the least-squares a posteriori error estimators, the solution can be accurately approximated even when the mesh is not aligned with discontinuity. The overshooting phenomenon is very mild if a piecewise constant reconstruction of the solution is used. Extensive numerical tests are done to show the effectiveness of the methods developed in the paper.

A true PML approach for steady-state vibration analysis of an elastically supported beam under moving load by a DLSFEM formulation

This paper concerns a computational implementation for solving a Moving Load (ML) problem on an infinite Euler–Bernoulli elastic beam on a Pasternak visco-elastic support. A steady-state dynamic response in convected coordinate is sought, by a numerical approach with discretization over a finite domain, implying spurious boundary reflections of non-evanescent waves. This is effectively solved by: (a) analytically formulating a new, true Perfectly Matched Layer (PML) approach, toward handling the underlying fourth-order differential problem and the corresponding far-field conditions, without adopting special boundary conditions; (b) outlining a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation, apt to provide a robust approach for the present non self-adjoint problem and to conveniently handle the jump condition in the shear force at the concentrated ML position. Consistent numerical results are illustrated and compared to an available analytical solution, showing a perfect match, with a complete removal of spurious boundary effects and a proof of theoretical a priori error estimates. Further results are produced for a case with multiple MLs. The paper shows that the present innovative DLSFEM-PML formulation is effectively suitable to numerically solve a steady-state ML problem on an infinite beam, setting up a new computational tool in such a challenging mechanical context.