Galerkin Formulation Research Articles

Convergence analysis of novel discontinuous Galerkin methods for a convection dominated problem

In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite element methods for the convection-dominated equation cause spurious oscillations. We choose to follow a DG finite element differential calculus framework introduced in Feng et al. (2016) and approximate the infinite-dimensional operators in the equation with the finite-dimensional DG differential operators. Specifically, we construct the numerical method by using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive term and the average discrete gradient operator for the convective term along with standard DG stabilization. We prove that the method converges optimally in the convection-dominated regime. Numerical results are provided to support the theoretical findings.

Computation of Deformable Interface Two‐Phase Flows: A Semi‐Lagrangian Finite Element Approach

ABSTRACTThis work aims at presenting a new computational approach to study two and three dimensional two‐phase flows and two dimensional coalescence phenomenon using direct numerical simulation. The flows are modeled by the incompressible Navier–Stokes equations, which are approximated by the finite element method. The Galerkin formulation is used to discretize the Navier–Stokes equations in the spatial domain and the semi‐Lagrangian method is used to discretize the material derivative. In order to satisfy the Ladyzhenskaya–Babuška–Brezzi condition, high‐order stable pair of elements are used, with pressure and velocity fields being calculated on different degrees of freedom in the unstructured mesh nodes. The interface is modeled by an unfitted adaptive moving mesh, where interface nodes are tracked in a Lagrangian fashion and moved with the velocity solution of the fluid motion equations. The surface tension is computed using the interface curvature and the gradient of a Heaviside function, and added in the momentum equations as a body force. In order to avoid undesired spurious modes at the interface due to high property ratios, a smooth transition between fluid properties is defined on the interface region. Several benchmark tests have been carried out to validate the proposed approach, and the obtained results have demonstrated agreement with analytical solutions and numerical results reported in the literature. A coalescence model is also proposed based on geometric criteria and results show interesting dynamics.

Meshless method for wave propagation in poroelastic transversely isotropic half‐space with the use of perfectly matched layer

AbstractNumerical investigation of wave propagation in transversely isotropic poroelastic half‐space with the use of a new stretched coordinate system through the Meshless Local Petrov–Galerkin (MLPG) formulation is presented in this paper. To this end, the u−p formulation of Biot is adopted as the framework of the porous media. One approach to numerically solve the infinite domain problems is the use of an absorber layer in which the whole half‐space is divided into two parts, that is (i) a finite part, in which the responses are interested, and (ii) the remaining semi‐infinite part, which is replaced by a Perfectly Matched Layer (PML). The stretched coordinates in the PML are introduced in such a way that the wave propagating in it does not generate spurious reflection to the finite part. Comparing the numerical results with some existing exact solutions and evaluating the norm of error demonstrate that the response functions in the finite part are achievable as precise as desired. Some new results are also presented which show the validity of the numerical approach in poroelastic transversely isotropic domain.

A computational study for simulating MHD duct flows at high Hartmann numbers using a stabilized finite element formulation with shock-capturing

The concern of this manuscript is stabilized finite element simulations of two-dimensional incompressible and viscous magnetohydrodynamic (MHD) duct flows governed by a coupled system of partial differential equations of the convection–diffusion type. In such flows, the high values of the Hartmann numbers (Ha) indicate that the convection process dominates the flow field. As is typical trouble encountered in convection-dominated flow simulations, classical discretization methods fail to function properly for MHD flows at high Hartmann numbers, yielding several numerical instabilities. In this study, the core computational tool we employ in order to overcome such challenges is the streamline-upwind/Petrov–Galerkin (SUPG) formulation. Beyond that, since the SUPG-stabilized formulation requires additional treatment where solutions experience rapid changes, we also enhance the SUPG-stabilized finite element formulation with a shock-capturing operator for achieving better solution profiles around sharp layers. The formulations are derived for both steady-state and time-dependent models. The proposed method, the so-called SUPG-YZβ formulation, techniques used, and in-house-developed finite element solvers are evaluated on a comprehensive set of numerical experiments. The test computations reveal that the SUPG-YZβ formulation yields quite good solution profiles near strong gradients without any significant numerical instabilities, even for quite challenging values of Ha, whereas the SUPG usually fails alone. Furthermore, by using only linear elements on relatively coarser meshes, the proposed formulation achieves this without the need for any adaptive mesh strategy, in contrast to most previously reported studies.

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

AbstractIn this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.

Energetic spectral-element time marching methods for phase-field nonlinear gradient systems

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

A unified discontinuous Galerkin formulation for interfacial multiphysics modeling of thermo-chemically driven fracture

Many engineering and natural materials exhibit coupled thermo-chemo-mechanical phenomena, which can result in embrittlement and fracture. These fractures, in turn, can alter the subsequent thermal, chemical, and mechanical response. We present a theoretical formulation and computational framework for the analysis of thermo-chemically fractured solids, with emphasis on the post-fracture thermal and chemical interfacial behavior. The theoretical model is based on the thermodynamically-consistent formulation of Loeffel and Anand (IJP, 2011). The computational method extends the scalable discontinuous Galerkin/Cohesive Zone Model (DG/CZM) of Radovitzky et al. (CMAME, 2011) to thermo-chemo-mechanics, which facilitates coupled, large-scale simulations of materials and structures containing failed interfaces. In the proposed framework, all balance laws are enforced weakly via the DG formalism, resulting in a unified formulation for multiphysics problems in solids. This naturally enables the incorporation of general interface models, e.g. to account for effects such as the aeolotropic reduction in thermochemical transport due to the presence of fractures, or the acceleration of chemical reactions along crack flanks. The approach is verified against two analytical solutions of boundary value problems drawn from thermo-poro-elasticity and thermally-driven delamination. A scalable, three-dimensional simulation of thermochemically-driven concrete cracking illustrates the complete capabilities of the interfacial multiphysics modeling framework.

Comparison of Physics Informed Neural Networks and Finite Element Method Solvers for advection-dominated diffusion problems

We present a comparison of Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN) with higher-order and continuity Finite Element Method (FEM). We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson–Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem on uniform grid. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov–Galerkin (PG) formulation and present the FEM solution obtained with the PG method. The main benefit of using a stabilization method is that it can deliver a good-quality approximation to the solution on a mesh that is not fully refined towards the singularity. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods. We consider a case with uniform FEM and uniform distribution of points for PINN, as well as uniform distribution of test functions for VPINN. We also consider adaptive FEM, refined towards edge singularity, and non-uniform distribution of points for PINN, as well as non-uniform distribution of test functions for VPINN.

A flux‐based HDG method

AbstractIn this article, we present a flux‐based formulation of the hybridizable discontinuous Galerkin (HDG) method for steady‐state diffusion problems and propose a new method derived by letting a stabilization parameter tend to infinity. Assuming an inf‐sup condition, we prove its well‐posedness and error estimates of optimal order. We show that the inf‐sup condition is satisfied by some triangular elements. Numerical results are also provided to support our theoretical results.

SUPG-based stabilized finite element computations of convection-dominated 3D elliptic PDEs using shock-capturing

This study is interested in stabilized finite element computations of advection-dominated convection–diffusion-type partial differential equations. The governing equations are assumed to be defined on the 3D unit cube and stationary. Towards that end, we stabilize the standard (Bubnov–) Galerkin finite element method (GFEM), which typically suffers from yielding numerical approximations polluted with node-to-node nonphysical oscillations in convection dominance, by employing the streamline-upwind/Petrov–Galerkin (SUPG) formulation. In order to achieve better shock representations near sharp layers, the SUPG-stabilized formulation is also enhanced with a simple and residual-based shock-capturing mechanism, the so-called YZβ technique. Several test computations are performed to assess the efficiency and robustness of the proposed formulation. The test problems are examined under more challenging conditions than those previously studied in the literature, i.e., for flows substantially dominated by convection phenomena. The numerical results and comparisons demonstrate that the SUPG-YZβ combination significantly eliminates both globally spread and localized numerical instabilities. Beyond that, the proposed formulation accomplishes that by using only linear elements on relatively coarser meshes and without making use of any adaptive mesh strategies.

High-order accurate transient and free-vibration analysis of plates and shells

The limited availability of analytical solutions and the high cost associated with experimental testing motivate the use of computational tools to assess the dynamic behavior of load-bearing components, especially when a wide design space must be explored, as is often the case with composite structures. In this context, a novel high-order accurate discontinuous Galerkin formulation for transient and free-vibration analysis of multilayered plates and shells is presented and numerical validated. The starting point of the formulation is a generalized structural theory for multilayered shells with arbitrary curvature based on the expansion of the displacement covariant components throughout the shell thickness. The variational statement of three-dimensional elastodynamics allows deriving the strong form of the governing differential equations, which form the basis to obtain the corresponding discontinuous Galerkin weak statements. As the order of the through-the-thickness expansion and the order of the discontinuous Galerkin basis functions are free parameters, the proposed approach allows tuning the order of accuracy of the computed solution throughout both the shell thickness and the shell modeling domain. Numerical results are reported and discussed for several validation test cases in terms of h- and p-convergence analyses, demonstrating the high-order accuracy, robustness, and computational savings of the formulation.

Multilevel Monte Carlo Methods for Stochastic Convection–Diffusion Eigenvalue Problems

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection–diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov–Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

A low-cost, penalty parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.

An efficient active-stress electromechanical isogeometric shell model for muscular thin film simulations

We propose an isogeometric approach to model the deformation of active thin films using layered, nonlinear, Kirchhoff–Love shells. Isogeometric Collocation and Galerkin formulations are employed to discretize the electrophysiological and mechanical sub-problems, respectively, with the possibility to adopt different element and time-step sizes.Numerical tests illustrate the capabilities of the active-stress-based approach to effectively simulate the contraction of thin films in both quasi-static and dynamic conditions.

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

Higher order phase-field modeling of brittle fracture via isogeometric analysis

AbstractThe evolution of brittle fracture in a material can be conveniently investigated by means of the phase-field technique introducing a smooth crack density functional. Following Borden et al. (2014), two distinct types of phase-field functional are considered: (i) a second-order model and (ii) a fourth-order one. The latter approach involves the bi-Laplacian of the phase field and therefore the resulting Galerkin form requires continuously differentiable basis functions: a condition we easily fulfill via Isogeometric Analysis. In this work, we provide an extensive comparison of the considered formulations performing several tests that progressively increase the complexity of the crack patterns. To measure the fracture length necessary in our accuracy evaluations, we propose an image-based algorithm that features an automatic skeletonization technique able to track complex fracture patterns. In all numerical results, damage irreversibility is handled in a straightforward and rigorous manner using the Projected Successive Over-Relaxation algorithm that is suitable to be adopted for both phase-field formulations since it can be used in combination with higher continuity isogeometric discretizations. Based on our results, the fourth-order approach provides higher rates of convergence and a greater accuracy. Moreover, we observe that fourth- and second-order models exhibit a comparable accuracy when the former methods employ a mesh-size approximately two times larger, entailing a substantial reduction of the computational effort.

A numerical investigation on coupling of conforming and hybridizable interior penalty discontinuous Galerkin methods for fractured groundwater flow problems

The present paper focuses on the numerical modeling of groundwater flows in fractured porous media using the codimensional model description. Therefore, fractures are defined explicitly as a (d−1)-dimensional geometric object immersed in a d-dimensional region and can act arbitrarily as a drain or a barrier. We numerically investigate a novel numerical strategy combining distinctive classes of conforming and nonconforming high-order Galerkin methods, both eligible for static condensation. This procedure is here particularly relevant, leading to a smaller and sparser final system with coupled degrees of freedom solely on the mesh skeleton. Precisely, we combine an inspired hybridizable interior penalty discontinuous Galerkin (HIP) formulation inside the bulk region and a standard continuous Galerkin (CG) approximation on the fracture network. The distinctive discretization of corresponding PDEs and the coupling strategy are rigorously exposed, and the local and global matrix assemblies are detailed. Extensive numerical experiments are then achieved to prove the model’s performances for 2D/3D analytic and realistic benchmarks. Qualitative comparisons are also considered with other discretization methods and commercial software, such as comsol.

Energy stable and structure-preserving schemes for the stochastic Galerkin shallow water equations

The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges that motivate the central advances of this paper. Starting with a spatially one-dimensional hyperbolicity-preserving, positivity-preserving stochastic Galerkin formulation of the parametric/uncertain shallow water equations, we derive an entropy-entropy flux pair for the system. We exploit this entropy-entropy flux pair to construct structure-preserving second-order energy conservative, and first- and second-order energy stable finite volume schemes for the stochastic Galerkin shallow water system. The performance of the methods is illustrated on several numerical experiments.

Adaptive Multi-level Algorithm for a Class of Nonlinear Problems

Abstract In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.

A fast numerical method for the conductivity of heterogeneous media with Dirichlet boundary conditions based on discrete sine–cosine transforms

The aim of this work is to develop a fast numerical method for solving conductivity problems in heterogeneous media subjected to Dirichlet boundary conditions. The method is based on a fixed-point iterative solution to an integral Lippmann–Schwinger type equation that is obtained by a Galerkin discretization of the cell problem using an approximation space spanned by sine series; the solution field is split between a known term verifying the Dirichlet boundary conditions and an unknown term described by sine series, which is null on the boundary by construction. With a suitable numerical integration scheme of the elementary integrals involved in the Galerkin formulation, based on discrete sine–cosine transforms, the method relies on the numerical complexity of fast Fourier transforms. The method is assessed in several problems including composite materials and fibrous networks.