Galerkin Least-squares Finite Element Method Research Articles

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.

Stabilized variational formulation of an oldroyd-B fluid flow equations on a Graphic Processing Unit (GPU) architecture

The governing equations of the flow of an oldroyd-B fluid are discretized using the finite element method. To overcome the convective nature of the momentum equation, the Galerkin/Least-Squares Finite Element Method (GLS/FEM) is used while the Discrete Elastic–Viscous Stress-Splitting (DEVSS) method is used to overcome the instability due to the absence of diffusion in the constitutive equations. The discretized equations are implemented on a hybrid system between the Graphics Processing Unit (GPU) architecture using Compute-Unified-Device-Architecture (CUDA) and a multi-core CPU. The implementation is applied successfully to simulate the blood flow in abdominal aortic aneurysm. To accelerate application performance on the GPU several optimized approaches are adopted. The most significant approach is the coloring technique that is used to assemble the global matrix. Numerical experiments show that the hybrid CPU-GPU implementation has a 26 time speedup over the multi-core CPU implementations.

Urban and overland flow modeling with dynamic adaptive mesh and implicit diffusive wave equation solver

A riverine and overland flow model based on the dynamically adapting Galerkin Least-Squares finite element method is developed and presented. The method utilizes the diffusive wave equations within a dynamically adaptive mesh and second-order accurate implicit time stepping scheme to efficiently and accurately simulate problems involving flows such as those that occur during overland flooding and urban inundation. The developed method and the utility of dynamic mesh adaption is presented through the simulation of common problems that occur in urban and overland flow conditions, these are shallow flooding, rapid flooding and flow propagation in an urban area. The application of the method to the flooding of the Morganza floodway illustrates the capabilities of the model in simulating large scale flooding and the application to a large dam break illustrates the limitations of the diffusive wave equations and the flow regimes where the numerical assumptions for the method break down. To the author's knowledge, the unique combination of FEM, wet-dry scheme and dynamic mesh adaption for the solution of diffusive wave equation (DWE) has not been presented in literature prior to this study.

A weak Galerkin least-squares finite element method for div–curl systems

In this paper, we introduce a weak Galerkin least-squares method for solving div–curl problem. This finite element method leads to a symmetric positive definite system and has the flexibility to work with general meshes such as hybrid mesh, polytopal mesh and mesh with hanging nodes. Error estimates of the finite element solution are derived. The numerical examples demonstrate the robustness and flexibility of the proposed method.

A Study on the Galerkin Least-Squares Method for the Oldroyd-B Model

Abstract We consider a reduced Galerkin least-squares finite element method for the Oldroyd-B model of viscoelastic fluid flows. Model problems considered are the flow past a planar channel and a 4-to-1 contraction problems. An a priori error estimate for the reduced Galerkin least-squares method is derived and numerical results supporting the estimate are presented.

A discontinuous Galerkin least-squares finite element method for solving coupled singularly perturbed reaction–diffusion equations

A discontinuous Galerkin least-squares finite element method is proposed to solve coupled reaction–diffusion equations with singular perturbations. This method produces solutions without numerical oscillations when uniform meshes are used. Numerical examples are provided to demonstrate the efficiency of the method.

A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling.

Haemodynamical simulations using one-dimensional (1D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. Recent interest in verifying 1D numerical schemes has led to the development of alternative experimental setups and the use of three-dimensional numerical models to acquire data not easily measured in vivo. In most studies to date, only one particular 1D scheme is tested. In this paper, we present a systematic comparison of six commonly used numerical schemes for 1D blood flow modelling: discontinuous Galerkin, locally conservative Galerkin, Galerkin least-squares finite element method, finite volume method, finite difference MacCormack method and a simplified trapezium rule method. Comparisons are made in a series of six benchmark test cases with an increasing degree of complexity. The accuracy of the numerical schemes is assessed by comparison with theoretical results, three-dimensional numerical data in compatible domains with distensible walls or experimental data in a network of silicone tubes. Results show a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries. All the information used in this study, including the input data for all benchmark cases, experimental data where available and numerical solutions for each scheme, is made publicly available online, providing a comprehensive reference data set to support the development of 1D models and numerical schemes.

A new absolutely stable simplified Galerkin Least-Squares finite element method using nonconforming element for the Stokes problem

We propose a new absolutely stable simplified Galerkin Least-Squares (SGLS) finite element method using equal order linear nonconforming element for the Stokes problem. We prove that our method is absolutely stable and has optimal error estimates. Numerical tests confirm and illustrate the theoretical analysis. Compare with the conforming Galerkin Least-Squares method and conforming SGLS method, our nonconforming method is more accuracy.

A simple stabilized finite element method for solving two phase compressible–incompressible interface flows

When computing interface flows between compressible (gas) and incompressible (liquid) fluids, one faces at least to the following difficulties: (1) transition from a gas density linked to the local temperature and pressure by an equation of state to a liquid density mainly constant in space, (2) proper approximation of the divergence constraint in incompressible regions and (3) wave transmission at the interface. The aim of the present paper is to design a global (i.e. the same for each phase) numerical method to address easily this coupling. To this end, the same set of primitive unknowns and equations is used everywhere in the flow, but with a dynamic parameterization that changes from compressible to incompressible regions. On one hand, the compressible Navier–Stokes equations are considered under weakly compressibility assumption so that a non-conservative formulation can be used. On the other hand, the incompressible non-isothermal model is retained. In addition, the level set transport equation is used to capture the interface position needed to identify the local characteristics of the fluid and to recover the adequate local modelling. For space approximation of Navier–Stokes equations, a Galerkin least-squares finite element method is used. Two essential elements for defining this numerical scheme are the stabilization and the computation of element integral of the approximated weak form. Since very different concerns motivate the need for stabilization in compressible and incompressible flows, the first difficulty is to design a stabilization operator suitable for both types of flows especially in mixed elements. In addition, some integral of discontinuous functions must be correctly computed to ensure interfacial wave transmission. To overcome these two difficulties, specific averages are computed especially near the interface. Finally, the level set transport equation is computed by a quadrature free Discontinuous Galerkin method. Numerical strategies are performed and validated for 1D and 2D applications.

Parallel Newton-Krylov solvers for modeling of a navigation lock filling system

The Galerkin least-squares finite element method for solving the Reynolds-averaged incompressible turbulent 3-D Navier-Stokes equations is employed to simulate a navigation lock filling system in the numerical code Adaptive Hydraulics (ADH). The linear system is solved at each nonlinear iteration within every time-step using biconjugate gradient stabilized (BiCGstab) in combination with block-Jacobi (bjacobi) preconditioners, as it failed to solve the linear system because of dramatic changes in flow velocity and pressure early in the simulation. To overcome this problem, we used the Portable Extensible Toolkit for Scientific Computation (PETSc), a numerical library that provides multiple types of linear solvers. PETSc has been incorporated into the ADH code. The ADH-PETSc interface helps to systematically investigate the best linear solver for an ADH simulation. We found that a variant, known as enhanced BiCGstab(l) in combination with the additive Schwarz method (ASM), made it possible to simulate the John Day lock filling system. The BiCGstab(l) solver improved the rate of convergence because of a more reliable update strategy for the residuals. In addition, the simulation was run with various numbers of processors. The result shows good scaling of solution time as the number of processors increases

Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities

In this paper, a discontinuous Galerkin least-squares finite element method is developed for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities by recasting the second-order elliptic equations as a system of first-order equations. In a companion paper (Lin in SIAM J Numer Anal 47:89–108, 2008) a similar method has been developed for problems with continuous data and shown to be well-posed, uniformly convergent, and optimal in convergence rate. In this paper the method is modified to take care of conditions that arise at interfaces and boundary singularities. Coercivity and uniform error estimates for the finite element approximation are established in an appropriately scaled norm. Numerical examples confirm the theoretical results.

Computation of Mean Drag for Bluff Body Problems Using Adaptive DNS/LES

We compute the time average of the drag in two benchmark bluff body problems: a surface mounted cube at Reynolds number 40000, and a square cylinder at Reynolds number 22000, using adaptive DNS/LES. In adaptive DNS/LES the Galerkin least-squares finite element method is used, with adaptive mesh refinement until a given stopping criterion is satisfied. Both the mesh refinement criterion and the stopping criterion are based on a posteriori error estimates of a given output of interest, in the form of a space-time integral of a computable residual multiplied by a dual weight, where the dual weight is obtained from solving an associated dual problem computationally, with the data of the dual problem coupling to the output of interest. No filtering is used, and in particular no Reynolds stresses are introduced. We thus circumvent the problem of closure, and instead we estimate the error contribution from subgrid modeling a posteriori, which we find to be small. We are able to predict the mean drag with an estimated tolerance of a few percent using about $10^5$ mesh points in space, with the computational power of a PC.

Hp -Discontinuous Galerkin Finite Element Methods with Least-Squares Stabilization

We consider a family of hp-version discontinuous Galerkin finite element methods with least-squares stabilization for symmetric systems of first-order partial differential equations. The family includes the classical discontinuous Galerkin finite element method, with and without streamline-diffusion stabilization, as well as the discontinuous version of the Galerkin least-squares finite element method. An hp-optimal error bound is derived in the associated DG-norm. If the solution of the problem is elementwise analytic, an exponential rate of convergence under p-refinement is proved. We perform numerical experiments both to illustrate the theoretical results and to compare the various methods within the family.