Continuous Galerkin Method Research Articles

A superconvergent postprocessing technique for the continuous Galerkin method in solving delay differential equations with nonlinear vanishing delay

A superconvergent postprocessing technique for the continuous Galerkin method in solving delay differential equations with nonlinear vanishing delay

Two Finite Element Approaches for the Porous Medium Equation That Are Positivity Preserving and Energy Stable

In this work, we present the construction of two distinct finite element approaches to solve the porous medium equation (PME). In the first approach, we transform the PME to a log-density variable formulation and construct a continuous Galerkin method. In the second approach, we introduce additional potential and velocity variables to rewrite the PME into a system of equations, for which we construct a mixed finite element method. Both approaches are first-order accurate, mass conserving, and proved to be unconditionally energy stable for their respective energies. The mixed approach is shown to preserve positivity under a CFL condition, while a much stronger property of unconditional bound preservation is proved for the log-density approach. A novel feature of our schemes is that they can handle compactly supported initial data without the need for any perturbation techniques. Furthermore, the log-density method can handle unstructured grids in any number of dimensions, while the mixed method can handle unstructured grids in two dimensions. We present results from several numerical experiments to demonstrate these properties.

Non-uniform knot (NUK) SIAC post-processing of flow fields produced through unstructured grid adaptation and optimization

As the finite element method (FEM) and the finite volume method (FVM), both their traditional and high-order variants, continue their proliferation into various applied engineering disciplines, adaptive mesh refinement and optimization strategies have increased in their importance when solving real-world computational fluid mechanics applications. The post-processing and visualization of the resulting flow fields present two significant analysis and visualization challenges. The first challenge is the handling of elemental continuity, which is often only C0 continuous (in continuous Galerkin methods) or piecewise discontinuous (in discontinuous Galerkin methods). The second challenge is that, depending on the flow regime and the geometric configurations for which adaptive meshing strategies are used, the meshes generated are often highly anisotropic. The (uniform knot) line-SIAC (L-SIAC) filter has been proposed as a way of handling elemental continuity issues in an accuracy-conserving manner with the added benefit of casting the data in a smooth context even if the representation is element discontinuous. In this paper, we demonstrate that the state-of-the-art adaptive L-SIAC filter, designed for mildly anisotropic meshes, suffers degradation in the quality of the post-processed solution when applied to the types of highly anisotropic meshes produced through adaptive mesh refinement and optimization. Hence, a new Non-Uniform Knot (NUK) L-SIAC filter is proposed that automatically conforms to the underlying mesh anisotropy. We demonstrate that the new filter behaves similarly to the adaptive L-SIAC filter when applied to uniform and mildly anisotropic meshes, and furthermore we show the superiority of the NUK L-SIAC filter when applied to highly anisotropic meshes. The newly formulated filter is applied to 2D canonical scalar fields and used to visualize 2D and 3D fluid flow simulation results.

Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods

We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral-element methods (DGSEMS). The use of Legendre-Gauss-Lobatto (LGL) quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain-preserving high-resolution schemes. Compared to many other continuous and discontinuous Galerkin method variants, a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcell flux discretization. Representing a high-order spatial semi-discretization in terms of intermediate states, we perform flux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains. In addition, local bounds may be imposed on scalar quantities of interest. In contrast to limiting approaches based on predictor-corrector algorithms, our MCL procedure for LGL-DGSEM yields nonlinear flux approximations that are independent of the time-step size and can be further modified to enforce entropy stability. To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations, we run simulations for challenging setups featuring strong shocks, steep density gradients, and vortex dominated flows.

On the enriched mixed Galerkin method with θ scheme for the elastic wave equation

In this paper, we propose the application of an enriched Galerkin (EG) method to solve the elastic wave equation in the “displacement pseudo-pressure” mixed form, which is known for being locking-free. Unlike the continuous Galerkin (CG) method and the discontinuous Galerkin (DG) method, the EG method is based on the weak form of the DG method, and the enriched space combines a continuous piecewise linear vector-valued function space with some discontinuous piecewise linear functions. Specifically, we employ the enriched space to discretize the displacement, while the pseudo-pressure is discretized using the P0 element space. As a result, compared to the DG method, the EG method significantly reduces the number of degrees of freedom. Furthermore, we adopt a more general θ-scheme for time discretization, which behaves as an implicit scheme for 14≤θ≤1 and as an explicit scheme for 0≤θ<14. We establish error estimates for both the semi-discrete and fully-discrete schemes. The accuracy of the proposed method is validated through numerical examples, which not only confirm the theoretical analysis but also provide simulations of elastic wave propagation.

Local conservation laws of continuous Galerkin method for the incompressible Navier–Stokes equations in EMAC form

We consider local balances of momentum and angular momentum for the incompressible Navier–Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier–Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory.

Numerical study of two-dimensional Burgers' equation by using a continuous Galerkin method

Numerical study of two-dimensional Burgers' equation by using a continuous Galerkin method

Numerical treatment of reactive diffusion using the discontinuous Galerkin method

AbstractThis work presents a new finite element variational formulation for the numerical treatment of diffusional phase transformations using the discontinuous Galerkin method (DGM). Steep concentration and property gradients near phase boundaries require particular focus on a sound numerical treatment. There are different ways to tackle this problem ranging from (i) the well-known phase field method (PFM) (Biner et al. in Programming phase-field modeling, Springer, Berlin, 2017, Emmerich in The diffuse interface approach in materials science: thermodynamic concepts and applications of phase-field models, Springer, Berlin, 2003), where the interface is described continuously to (ii) methods that allow sharp transitions at phase boundaries, such as reactive diffusion models (Svoboda and Fischer in Comput Mater Sci 127:136–140, 2017, 78:39–46, 2013, Svoboda et al. in Comput Mater Sci 95:309–315, 2014). Phase transformation problems with continuous property changes can be implemented using the continuous Galerkin method (GM). Sharp interface models, however, lead to stability problems with the GM. A method that is able to treat the features of sharp interface models is the discontinuous Galerkin method. This method is well understood for regular diffusion problems (Cockburn in ZAMM J Appl Math Mech 83(11):731–754, 2003). As will be shown, it is also particularly well suited to model phase transformations. We discuss the thermodynamic background by review of a multi-phase, binary system. A new DGM formulation for the phase transformation problem with sharp interfaces is then introduced. Finally, the derived method is used in a 2D microstructural evolution simulation that features a binary, three-phase system that also takes the vacancy mechanism of solid body diffusion into account.

Numerical analysis of a space-time continuous Galerkin method for the nonlinear Sobolev equation

In this work, we shall use space-time continuous Galerkin (STCG) method to analyze the nonlinear Sobolev equation. This method is different from the conventional finite element methods, that is, it not only uses finite element to discrete spatial variables but also uses finite element to discrete temporal variable. Therefore it is easier to obtain high order accuracy in time and space than the conventional finite element methods and has the good stability. Also, it does not need time step size and space mesh parameter to satisfy the stability conditions. In this paper, we show a detailed analysis including the proof of the existence and uniqueness of the numerical solution and the a priori error estimate. Especially, the analysis method given here can be similarly applied to the more general nonlinear problems. At last, the numerical examples reveal that the iterative method is more efficient than the linearized method for the nonlinear problems with STCG method. Also, the numerical experiments show that the STCG method is more efficient than the space-time discontinuous Galerkin (STDG) method in the practical calculations.

Fully coupled hydro-mechanical modeling of two-phase flow in deformable fractured porous media with discontinuous and continuous Galerkin method

Discrete fracture and matrix modeling of coupled flow and geomechanics is instrumental for understanding flow in fractured media for various geoengineering applications such as enhanced geothermal energy systems and groundwater remediation. We employ the governing equations for two-phase flow in deformable fractured porous media with a stress-dependent porosity and permeability model of matrix, and variable fracture aperture and the corresponding permeability. A finite element framework is presented, in which fractures are regarded as low-dimensional objects, to discretize the coupled two-phase flow and geomechanics in fractured porous media. A hybrid method, combining discontinuous Galerkin (DG) and continuous Galerkin (CG) finite element methods (FEM), is utilized to solve for the two-phase flow, while the solid deformation is approximated using a discontinuous Galerkin FEM approach. Several benchmark cases are utilized to examine the accuracy of the proposed hybrid DG-CG FEM method. Further validation is performed using more complex, realistic fracture configurations. As the mechanical characteristics of the fracture and the surrounding matrix differ, the simulation results demonstrate that displacement and stress are discontinuous on both sides of the fracture. While the case with low permeability fractures exhibits pressure jump across the fractures, the pressure changes are reasonably smooth for the conduit fractures.

An adaptive viscosity regularization approach for the numerical solution of conservation laws: Application to finite element methods

We introduce an adaptive viscosity regularization approach for the numerical solution of systems of nonlinear conservation laws with shock waves. The approach seeks to solve a sequence of regularized problems consisting of the system of conservation laws and an additional Helmholtz equation for the artificial viscosity. We propose a homotopy continuation of the regularization parameters to minimize the amount of artificial viscosity subject to positivity-preserving and smoothness constraints on the numerical solution. The regularization methodology is combined with a mesh adaptation strategy that identifies the shock location and generates shock-aligned meshes, which allows to further reduce the amount of artificial dissipation and capture shocks with increased accuracy. We use the hybridizable discontinuous Galerkin method to numerically solve the regularized system of conservation laws and the continuous Galerkin method to solve the Helmholtz equation for the artificial viscosity. We show that the approach can produce approximate solutions that converge to the exact solution of the Burgers' equation. Finally, we demonstrate the performance of the method on inviscid transonic, supersonic, hypersonic flows in two dimensions. The approach is found to be accurate, robust and efficient, and yields very sharp yet smooth solutions in a few homotopy iterations.

Efficient numerical solution of the Fokker–Planck equation using physics-conforming finite element methods

Abstract We consider the Fokker–Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternating-direction methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier–Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system.

Feedback control for fluid mixing via advection

This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain Ω⊂Rd, with d=2 or d=3. The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space (H1(Ω))′ of H1(Ω) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws.

An Adaptive Model Reduction Method Leveraging Locally Supported Basis Functions

We propose a new method, the continuous Galerkin method with globally and locally supported basis functions (CG-GL), to address the parametric robustness issues of reduced-order models (ROMs) by incorporating solution-based adaptivity with locally supported finite element basis functions. The CG-GL method combines the accuracy of locally supported basis functions with the efficiency of globally supported data-driven basis functions. Efficient output-based dual-weighted residual error estimates are derived and implemented for the CG-GL method and used to drive efficient online trial space adaptation. An empirical quadrature procedure is introduced for rapid evaluation of nonlinear terms that does not require retraining throughout the adaptation process. Two numerical experiments demonstrate the potential of the CG-GL method to produce accurate approximations with limited training and its tunable trade-off between accuracy and computational cost.

Superconvergence and postprocessing of the continuous Galerkin method for nonlinear Volterra integro-differential equations

We propose a novel postprocessing technique for improving the global accuracy of the continuous Galerkin (CG) method for nonlinear Volterra integro-differential equations. The key idea behind the postprocessing technique is to add a higher order Lobatto polynomial of degree k + 1 to the CG approximation of degree k. We first show that the CG method superconverges at the nodal points of the time partition. We further prove that the postprocessed CG approximation converges one order faster than the unprocessed CG approximation in the L2-, H1- and L∞-norms. As a by-product of the postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact. Numerical examples are presented to highlight the superconvergence properties of the postprocessed CG approximations and the robustness of the a posteriori error estimators.

A Non-perturbative Approach to Computing Seismic Normal Modes in Rotating Planets

A continuous Galerkin method based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard “radial” algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales.

An assessment of solvers for algebraically stabilized discretizations of convection–diffusion–reaction equations

Abstract We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

The software design of Gridap: A Finite Element package based on the Julia JIT compiler

We present the software design of Gridap, a novel finite element library written exclusively in the Julia programming language, which is being used by several research groups world-wide to simulate complex physical phenomena such as magnetohydrodynamics, photonics, weather modeling, non-linear solid mechanics, and fluid-structure interaction problems. The library provides a feature-rich set of discretization techniques for the numerical approximation of a wide range of mathematical models governed by Partial Differential Equations (PDEs), including linear, nonlinear, single-field, and multi-field equations. An expressive API allows users to define PDEs in weak form by a syntax close to the mathematical notation. While this is also available in previous frameworks, the main novelty of Gridap is that it implements this API without introducing a domain-specific language plus a compiler of variational forms. Instead, it leverages the Julia just-in-time compiler to build efficient code, specialized for the concrete problem at hand. As a result, there is no need to use different languages for the computational back-end and the user front-end anymore, thus eliminating the so-called two-language problem. Gridap also provides a low-level API that is modular and extensible via the multiple-dispatch paradigm of Julia and provides easy access to the main building blocks of the library if required. The main contribution of this paper is the detailed presentation of the novel software abstractions behind the Gridap design that leverages the new software possibilities provided by the Julia language. The second main contribution of the article is a performance comparison against FEniCS. We measure CPU times needed to assemble discrete systems of linear equations for different problem types and show that the performance of Gridap is comparable to FEniCS, demonstrating that the new software design does not compromise performance. Gridap is freely available at Github (github.com/gridap/Gridap.jl) and distributed under an MIT license. Program summaryProgram title: Gridap.jl (version 0.16)CPC Library link to program files:https://doi.org/10.17632/mh9vv7hrgf.1Developer's repository link:https://github.com/gridap/Gridap.jlLicensing provisions: MIT licenseProgramming language: JuliaSupplementary material: Source code of the Listings presented in this paper. Each Listing below indicates the name of its corresponding source file.Nature of problem: Computational simulation of a broad range of application problems governed by partial differential equations including linear, nonlinear, single field, and multi-physics problems. Gridap is currently being used by several research groups world-wide to simulate complex physical phenomena such as magnetohydrodynamics, photonics, weather modeling, non-linear solid mechanics, and fluid-structure interaction problems.Solution method: Arbitrary-order grad-, curl-, and div-conforming finite elements on n-cube and n-simplex meshes. Continuous and Discontinuous Galerkin methods. Newton-Raphson linearization. Krylov subspace iterative solvers. Sparse direct solvers.

Discontinuous and continuous Galerkin methods for compressible single-phase and two-phase flow in fractured porous media

Accurate simulation of flow behaviors in fractured porous media is challenging. We present a discontinuous Galerkin (DG) approximation and continuous Galerkin (CG) approximation for compressible single- and two-phase flow in porous media with conducting (high permeable) and blocking (low permeable) fractures using a mixed-dimensional approach in which the fracture is described as a reduced-dimensional interface coupled with linear transmission conditions. The proposed DG/CG method was first verified with single-phase fractured flow benchmark cases and then applied to time-dependent single-phase flow cases. The simulated results demonstrate that the DG/CG method is capable of capturing the continuity as well as the jump in pressure between the two sides of the matrix-fracture interface. For the two-phase flow cases, we verify the DG/CG method with a reference case involving a complex fracture configuration. Subsequently, we analyze several cases to study two-phase flow through a single fracture and a discrete fracture network in two dimensions. Overall, the simulation results show that the developed DG/CG approach can reliably predict the flow behaviors for single- and two-phase flow in fractured porous media.

The comparison of results obtained from the continuous and discontinuous Galerkin Method for the thermoelasticity problem

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology