Least-squares Finite Element Method Research Articles

An Adaptive Nonlinear Least-Squares Finite Element Method for a Pucci Equation in Two Dimensions

An Adaptive Nonlinear Least-Squares Finite Element Method for a Pucci Equation in Two Dimensions

Stabilized equal lower-order finite element methods for simulating Brinkman equations in porous media

This paper demonstrates the mixed formulation of the Brinkman problem using linear equal-order finite element methods in porous media modelling. We introduce Galerkin least-squares (GLS) and least-squares (LS) finite element methods to address the incompatibility of finite element spaces, treating velocity, pressure, and vorticity as independent variables. Theoretical analysis examines coercivity and continuity, providing error estimates. Demonstrating resilience in theoretical findings, these methods achieve optimal convergence rates in the L 2 norm by incorporating stabilization terms with low-order basis functions. Numerical experiments validate theoretical predictions, showing the effectiveness of the GLS method and addressing finite element space incompatibility. Additionally, the GLS method exhibits promising capabilities in handling the Brinkman equation at low permeability compared to the LS method. The study reveals an increase in the average pressure difference in the Brinkman problem compared to the Stokes equations as the inlet velocity rises, providing insights into the behaviour of Brinkman equations.

Adaptive least-squares finite element methods: Guaranteed upper bounds and convergence in L2 norm of the dual variable

We consider adaptive least-squares finite element methods. First, we develop a guaranteed upper bound for the dual error in the L2 norm, and this can be used as a stopping criterion for the adaptive procedures. Secondly, based on the a posteriori error estimates for the dual variable, we develop an error indicator that identifies the local area to refine, and establish the convergence of the adaptive procedures based on the Dörfler's marking strategy. Our convergence analysis is valid for the entire range of the bulk parameter 0<Θ≤1 and it shows the effect of bulk parameter and reduction factor of elements on the convergence rate. Confirming numerical experiments are provided.

A least squares finite element method for time fractional telegraph equation with Vieta-Lucas basis functions

We proposed a Least Squares Finite Element Method (LSFEM) for the approximate solution of Time fractional telegraph equation (TFTE). The method implemented the Vieta-Lucas polynomials expressed in the Caputo fractional sense as basis functions. The LSFEM reduces the TFTE into algebraic system with element matrices and vectors, while the Conjugate Gradient Method (CGM) solved the system for the unknowns with the aid of MAPLE 18. The results generated implies that the proposed method converges favorably to the theoretical solution when the number of elements is large with a large space-time step used for the parameter t.

Simulation of SN transport equation for hexagonal-z reactor using the COMSOL Multiphysics software

Detailed neutron transport simulation has undergone much investigation in advanced reactor engineering, where the adaptability of complex geometrics and the simplicity of computational implementation are of great significance. In this paper, a detailed neutron transport module (DNTM) based on the COMSOL Multiphysics platform is developed for hexagonal-z reactor conditions. The multi-group, multidimensional neutron transport SN equation is discretized using the least-squares finite element method and is embedded in COMSOL using the weak form partial differential equation interface without the necessity for additional coding. The proposed DNTM’s feasibility and accuracy are demonstrated through the analysis of three neutron transport benchmarks, encompassing various energy groups, dimensions, and scales. All the numerical results show good agreement with the reference values, indicating that the proposed DNTM in COMSOL can be used to solve the neutron transport process in the hexagonal-z reactor accurately and reliably. This approach may contribute a simple and practical high-fidelity neutronics solving tool and an effective direction to realize multi-physics tight coupling simulations.

A mixed least‐squares finite element method for a geometrically nonlinear TPM formulation

AbstractIn the present work, a mixed least‐squares finite element method (LSFEM) is used, which is applied on the partial differential equations arising from the Theory of Porous Media (TPM), see for example [1, 2, 3]. Since the LSFEM is not limited to the LBB condition, the method has some theoretical advantages over the well‐known (mixed) Galerkin method, see [4]. In particular, the LSFEM leads to positive definite and symmetric system matrices, even for differential equations with non‐self‐adjoint operators. In detail, we study an incompressible binary model with solid and liquid phases. The modeling of saturated porous structures serves as the basis for the presented approach. The resulting finite element is a four‐field formulation in terms of solid displacements, fluid pressure, mixture stresses, and a new variable associated with the pressure gradient. Vector‐valued Raviart‐Thomas and conventional Lagrangian functions are used to achieve a conforming discretization of the unknowns in the spaces H(div) and H1. The applicability of the LSFEM approach is demonstrated with numerical examples for fluid‐saturated porous structures, which consider incompressible, geometrically nonlinear elastic material behavior.

A Simulation Study on von Karman Vortex Shedding with Navier-Stokes and Shallow-Water Models

This study aims to investigate the advantages of employing numerical models based on Shallow-water equations for simulating von Karman vortex shedding. Furthermore, a comparative analysis with Navier-Stokes equations will be conducted to assess their effectiveness. In addition to Reynolds number (Re), Froude number (Fr), relevant to water depth, plays an important role in the Shallow-Water modeling of the von Karman vortex. In this study, simulations of 2D von Karman vortex shedding are performed using the Navier-Stokes model and Shallow-Water model, employing the least-squares finite-element method for space discretization and θ-method for time integration. The computed vortices characteristics, including the recirculation zone behind the cylinder, vortices size, and frequency, are presented. In the Navier-Stokes modeling, the computed results indicate that the size of vortices in space decreases and the Strouhal number increases as Re increases. In the Shallow-Water modeling for the same Re condition, the size of vortices increases and the Strouhal number decreases as Fr increases.

First-order system least-squares finite element method for singularly perturbed Darcy equations

We define and analyse a least-squares finite element method for a first-order reformulation of a scaled Brinkman model of fluid flow through porous media. We introduce a pseudostress variable that allows to eliminate the pressure variable from the system. It can be recovered by a simple post-processing. It is shown that the least-squares functional is uniformly equivalent, i.e., independent of the singular perturbation parameter, to a parameter dependent norm. This norm equivalence implies that the least-squares functional evaluated in the discrete solution provides an efficient and reliable a posteriori error estimator. Numerical experiments are presented.

Least-Squares Methods with Nonconforming Finite Elements for General Second-Order Elliptic Equations

In this paper, we study least-squares finite element methods (LSFEM) for general second-order elliptic equations with nonconforming finite element approximations. The equation may be indefinite. For the two-field potential-flux div LSFEM with Crouzeix–Raviart (CR) element approximation, we present three proofs of the discrete solvability under the condition that mesh size is small enough. One of the proof is based on the coerciveness of the original bilinear form. The other two are based on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. A counterexample shows that div least-squares functional does not have norm equivalence in the sum space of $$H^1$$ and CR finite element spaces. Thus it cannot be used as an a posteriori error estimator. Several versions of reliable and efficient error estimators are proposed for the method. We also propose a three-filed potential-flux-intensity div-curl least-squares method with general nonconforming finite element approximations. The norm equivalence in the abstract nonconforming piecewise $$H^1$$ -space is established for the three-filed formulation on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. The three-filed div-curl nonconforming formulation thus has no restriction on the mesh size, and the least-squares functional can be used as the built-in a posteriori error estimator. Under some restrictive conditions, we also discuss a potential-flux div-curl least-squares method.

Least-squares finite element method for ordinary differential equations

We consider the least-squares finite element method (lsfem) for systems of nonlinear ordinary differential equations, and establish an optimal error estimate for this method when piecewise linear elements are used. The main assumptions are that the vector field is sufficiently smooth and that the local Lipschitz constant as well as the operator norm of the Jacobian matrix associated with the nonlinearity are sufficiently small, when restricted to a suitable neighborhood of the true solution for the considered initial value problem. This theoretic optimality is further illustrated numerically, along with evidence of possible extension to higher-order basis elements. Examples are also presented to show the advantages of lsfem compared with finite difference methods in various scenarios. Suitable modifications for adaptive time-stepping are discussed as well.

Down-Hole Electromagnetic Heating of Deep Aquifers for Renewable Energy Storage

Electromagnetic (EM) heating is an emerging method for storing renewable energy, such as photovoltaic solar and wind electric power, into aquifers. We investigate how the captured energy increases the temperature of a prototypical deep aquifer for a six-month period and then to which extent the stored energy can be recovered during the consecutive six months. Water injected at a constant flow rate is simultaneously heated using a high-frequency electromagnetic microwave emitter operating at the water natural resonance frequency of 2.45 GHz. The coupled reservoir flow and EM heating are described using Darcy’s and the energy balance equations. The latter includes a source term accounting for the EM wave propagation and absorption, modeled separately using Maxwell’s equations. The equations are solved numerically by the Galerkin least-squares finite element method. The approach was validated using EM-heating input data obtained from controlled laboratory experiments and then was applied to the aquifer. We found that after six years of alternate storage and recovery, up to 77% of the injected energy is recovered when considering realistic heat losses estimated from field data. Even when heat losses are increased by a factor of two, up to 69% of the injected energy is recovered in this case. This shows that down-hole EM heating is a highly effective method for storing renewable energies, capable of helping to solve their inherent intermittency.

MINRES for Second-Order PDEs with Singular Data

Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov--Galerkin method with optimal test functions (DPG) usually exclude singular data, e.g., non square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of $H^{-1}$ loads, and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.

A finite element level-set method for stress-based topology optimization of plate structures

In this paper, we present a stress-based topology optimization of plate structures by using a finite element level-set (LS) method. Since stress constraints are independent with LS-based design variables, the singularity phenomenon arising in topology optimization under stress constraints can be resolved without using any relaxation technique. The finite element method is used for solving LS-based equations of design variables and the state and adjoint equations of plate structures with arbitrary complex geometries and boundaries of the design domain. To obtain stable finite element solutions without any oscillation, both unsteady LS convection-diffusion equation and LS re-initialization convection-reaction equation are solved using a stabilized least-squares finite element method.

How to prove optimal convergence rates for adaptive least-squares finite element methods∗

Abstract The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53 (1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions.

Least-Squares Finite Element Method for Solving Stokes Flow under Point Source Magnetic Field

The least-squares finite element method (LSFEM) is successfully employed for the discretization of the Stokes equations and the numerical computation of the behaviour of two-dimensional Stokes flow in a straight rectangular channel under the effect of a point-source magnetic field. LSFEM has several advantages in terms of theory and computing, where it can always create a symmetric, positive-definite algebraic system of equations. It also allows for using an equal order shape function for both velocity and pressure, and it is not required to satisfy the Ladyzhenskaya–Babuška–Brezzi (LBB) condition. Despite this, LSFEM has an issue where low-order nodal expansions tend to lock. Thus, the present study proposes the discretization of the problem domain using higher-order nodes elements with full numerical integration. Results concerning velocity contour and streamlines pattern are shown. On the basis of current findings, it can be concluded that the LSFEM can be used to solve Stokes flow problem under the point source magnetic field.

3D modeling of generalized Newtonian fluid flow with data assimilation using the least-squares finite element method

In this contribution we present a least-squares finite element formulation to model steady-state flow of incompressible, non-Newtonian fluids in three dimensions including data assimilation. The approach is based on the incompressible Navier–Stokes equations and the nonlinear viscosity is considered by means of the Carreau–Yasuda viscosity model, which can account for shear-thickening and shear-thinning behavior of generalized Newtonian fluids. We outline the procedure how to integrate given data into the numerical solution of flow problems without additional computational cost using the least-squares FEM. Assimilation of experimental data provides the opportunity to reduce model errors resulting in a solution which more closely approximates reality. Furthermore, the preprocessing of the available data using Kriging interpolation is also described briefly.The presented formulation is first validated by investigating the flow in a cube with an exact solution without data assimilation. Convergence is evaluated based on the error in velocities and pressure compared to the exact solution. Then the effect of data assimilation is shown by modeling blood flow through a carotid bifurcation model and integrating data either along lines or over entire cross-sectional areas. The improvement of the numerical solution by means of data assimilation is revealed by comparing the calculated velocity profiles with experimental and numerical reference values.

Solution of Shallow-Water Equations by a Layer-Integrated Hydrostatic Least-Squares Finite-Element Method

A multi-layer hydrostatic shallow-water model was developed in the present study. The layer-integrated hydrostatic nonlinear shallow-water was solved with θ time integration and the least-squares finite element method. Since the least-squares formulation was employed, the resulting system of equations was symmetric and positive–definite; therefore, it could be solved efficiently by the preconditioned conjugate gradient method. The model was first applied to simulate the von Karman vortex shedding. A well-organized von Karman vortex street was reproduced. The model was then applied to simulate the Kuroshio current-induced Green Island vortex street. A swirling recirculation was formed and followed by several pairs of alternating counter-rotating vortices. The size of the recirculation, as well as the temporal and spatial scale of the vortex shedding, were found to be consistent with ADCP-CDT measurements, X-band radar measurements, and analysis of the satellite images. It was also revealed that Green Island vortices were affected by the upstream Orchid Island vortices.

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].