Continuous Galerkin Method Research Articles

Generalized multiscale finite element method for elasticity equations

In this paper, we discuss the application of generalized multiscale finite element method (GMsFEM) to elasticity equation in heterogeneous media. We consider steady state elasticity equations though some of our applications are motivated by elastic wave propagation in subsurface where the subsurface properties can be highly heterogeneous and have high contrast. We present the construction of main ingredients for GMsFEM such as the snapshot space and offline spaces. The latter is constructed using local spectral decomposition in the snapshot space. The spectral decomposition is based on the analysis which is provided in the paper. We consider both continuous Galerkin and discontinuous Galerkin coupling of basis functions. Both approaches have their cons and pros. Continuous Galerkin methods allow avoiding penalty parameters though they involve partition of unity functions which can alter the properties of multiscale basis functions. On the other hand, discontinuous Galerkin techniques allow gluing multiscale basis functions without any modifications. Because basis functions are constructed independently from each other, this approach provides an advantage. We discuss the use of oversampling techniques that use snapshots in larger regions to construct the offline space. We provide numerical results to show that one can accurately approximate the solution using reduced number of degrees of freedom.

Comparison of four different obstacle models of fluid flow with a slip-like boundary condition

In this paper, we investigate a time-discretized 2-dimensional Navier-Stokes equation with a slip-like boundary condition, which arises in the melting ice problem with obstacle. We study the existence and uniqueness of a approximate solution. We also study the numerical solution of melting ice problem using Continuous Galerkin method.

A posteriori error estimates for continuous interior penalty Galerkin approximation of transient convection diffusion optimal control problems

In this paper a posteriori error estimate for continuous interior penalty Galerkin approximation of transient convection dominated diffusion optimal control problems with control constraints is presented. The state equation is discretized by the continuous interior penalty Galerkin method with continuous piecewise linear polynomial space and the control variable is approximated by implicit discretization concept. By use of the elliptic reconstruction technique proposed for parabolic equations, a posteriori error estimates for state variable, adjoint state variable and control variable are proved, which can be used to guide the mesh refinement in the adaptive algorithm.

A hybridizable discontinuous Galerkin formulation for non-linear elasticity

We revisit the hybridizable discontinuous Galerkin method for non-linear elasticity introduced by S.-C. Soon (2008). We show that it can be recast as a minimization problem of a non-linear functional over a space of discontinuous approximations to the displacement. The functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter-element jumps of the displacement. We then show that if this new energy is not properly weighted, the minimizers might not converge to the exact solution. We construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter-element jumps. Finally, we explore the performance of the method for the case of piecewise-linear approximations in rather demanding situations in both two-dimensional and, for the first time, three-dimensional situations. They include almost incompressible materials, large deformations with large-shear layers, and cavitation. We also compare the method with the continuous Galerkin method and a previously explored discontinuous Galerkin method, and show that, when using piecewise-linear approximations and a moderate number of degrees of freedom, the current method turns out to be more efficient for the computation of the gradient.

Variational space–time (dis)continuous Galerkin method for nonlinear free surface water waves

A new variational finite element method is developed for nonlinear free surface gravity water waves using the potential flow approximation. This method also handles waves generated by a wave maker. Its formulation stems from Miles' variational principle for water waves together with a finite element discretization that is continuous in space and discontinuous in time. One novel feature of this variational finite element approach is that the free surface evolution is variationally dependent on the mesh deformation vis-à-vis the mesh deformation being geometrically dependent on free surface evolution. Another key feature is the use of a variational (dis)continuous Galerkin finite element discretization in time. Moreover, in the absence of a wave maker, it is shown to be equivalent to the second order symplectic Störmer–Verlet time stepping scheme for the free-surface degrees of freedom. These key features add to the stability of the numerical method. Finally, the resulting numerical scheme is verified against nonlinear analytical solutions with long time simulations and validated against experimental measurements of driven wave solutions in a wave basin of the Maritime Research Institute Netherlands.

Variational time discretization methods for optimal control problems governed by diffusion–convection–reaction equations

In this paper, the distributed optimal control problem governed by unsteady diffusion–convection–reaction equation without control constraints is studied. Time discretization is performed by variational discretization using continuous and discontinuous Galerkin methods, while symmetric interior penalty Galerkin with upwinding is used for space discretization. We investigate the commutativity properties of the optimize-then-discretize and discretize-then-optimize approaches for the continuous and discontinuous Galerkin time discretization. A priori error estimates are derived for fully-discrete state, adjoint and control. The numerical results given for convection dominated problems via optimize-then-discretize approach confirm the theoretically observed convergence rates.

Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method

We describe an algorithm for modeling saturated fractures in a poroelastic domain in which the reservoir simulator is coupled with a boundary element method. A fixed stress splitting is used on the underlying fractured Biot system to iteratively couple fluid and solid mechanics systems. The fluid system consists of Darcy’s law in the reservoir and is computed with a multipoint flux mixed finite element method, and a Reynolds’ lubrication equation in the fracture solved with a mimetic finite difference method. The mechanics system consists of linear elasticity in the reservoir and is computed with a continuous Galerkin method, and linear elasticity in the fracture is solved with a weakly singular symmetric Galerkin boundary element method. This algorithm is able to compute both unknown fracture width and unknown fluid leakage rate. An interesting numerical example is presented with an injection well inside of a circular fracture.

High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling.

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

Numerical convergence study of iterative coupling for coupled flow and geomechanics

In this paper, we consider algorithms for modeling complex processes in porous media that include fluid and structure interactions. Numerous field applications would benefit from a better understanding and integration of porous flow and solid deformation. Important applications in environmental and petroleum engineering include carbon sequestration, surface subsidence, pore collapse, cavity generation, hydraulic fracturing, thermal fracturing, wellbore collapse, sand production, fault activation, and waste disposal, while similar issues arise in biosciences and chemical sciences as well. Here, we consider solving iteratively the coupling of flow and mechanics. We employ mixed finite element method for flow and a continuous Galerkin method for elasticity. For single-phase flow, we demonstrate the convergence and convergence rates for two widely used schemes, the undrained split and the fixed stress split. We discuss the extension of the fixed stress iterative coupling scheme to an equation of state compositional flow model coupled with elasticity and a single-phase poroelasticity model on general hexahedral grids. Computational results are presented.

Approaches for Adjoint-Based A Posteriori Analysis of Stabilized Finite Element Methods

This study derives a posteriori error estimates for linear functionals of the solution of systems of partial differential equations discretized using stabilized continuous Galerkin methods. We investigate a convection-diffusion equation, the Stokes equations, and incompressible low Reynolds number flow governed by the Navier--Stokes equations. We consider three well-known stabilization methods and show that only one of the three is adjoint consistent and that even this case is contingent upon a proper treatment of the adjoint data. A standard approach for a posteriori error analysis uses the adjoint of the stabilized formulation, which inherits the difficulties induced by the lack of adjoint consistency. We introduce and analyze two alternative approaches. The first is based on the addition of stabilization terms to the adjoint data, while the second is based on a stabilized formulation of the formal adjoint problem. We show that any of the three approaches can be used to derive a fully computable error representation. However, numerical results show that these new alternative approaches result in more accurate error estimates for a variety of problems.

Coupling multipoint flux mixed finite element methodswith continuous Galerkin methods for poroelasticity

We study the numerical approximation on irregular domains with general grids of the system of poroelasticity, which describes fluid flow in deformable porous media. The flow equation is discretized by a multipoint flux mixed finite element method and the displacements are approximated by a continuous Galerkin finite element method. First-order convergence in space and time is established in appropriate norms for the pressure, velocity, and displacement. Numerical results are presented that illustrate the behavior of the method.

BDDC preconditioners for continuous and discontinuous Galerkin methods using spectral/hp elements with variable local polynomial degree

Locally adapted meshes and polynomial degrees can greatly improve spectral element accuracy and applicability. A balancing domain decomposition by constraints (BDDC) preconditioner is constructed and analysed for both continuous (CG) and discontinuous (DG) Galerkin discretizations of scalar elliptic problems, built by nodal spectral elements with variable polynomial degrees. The DG case is reduced to the CG case via the auxiliary space method. The proposed BDDC preconditioner is proved to be scalable in the number of subdomains and quasi-optimal in both the ratio of local polynomial degrees and element sizes and the ratio of subdomain and element sizes. Several numerical experiments in the plane confirm the obtained theoretical convergence rate estimates, and illustrate the preconditioner performance for both CG and DG discretizations. Different configurations with locally adapted polynomial degrees are studied, as well as the preconditioner robustness with respect to discontinuities of the elliptic coefficients across subdomain boundaries. These results apply also to other dual-primal preconditioners defined by the same set of primal constraints, such as FETI-DP preconditioners.

Efficiency of high‐order elements for continuous and discontinuous Galerkin methods

SUMMARYTo evaluate the computational performance of high‐order elements, a comparison based on operation count is proposed instead of runtime comparisons. More specifically, linear versus high‐order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution. Continuous and discontinuous Galerkin methods are considered in two‐dimensional and three‐dimensional domains for simplices and parallelotopes. Moreover, both element‐wise and global operations arising from different Galerkin approaches are studied. The operation count estimates show, that for implicit solvers, high‐order methods are more efficient than linear ones. Copyright © 2013 John Wiley & Sons, Ltd.

High‐order continuous and discontinuous Galerkin methods for wave problems

SUMMARYThree Galerkin methods—continuous Galerkin, Compact Discontinuous Galerkin, and hybridizable discontinuous Galerkin—are compared in terms of performance and computational efficiency in 2‐D scattering problems for low and high‐order polynomial approximations. The total number of DOFs and the total runtime are used for this correlation as well as the corresponding precision. The comparison is carried out through various numerical examples. The superior performance of high‐order elements is shown. At the same time, similar capabilities are shown for continuous Galerkin and hybridizable discontinuous Galerkin, when high‐order elements are adopted, both of them clearly outperforming compact discontinuous Galerkin. Copyright © 2013 John Wiley & Sons, Ltd.

Discretization of the Wave Equation Using Continuous Elements in Time and a Hybridizable Discontinuous Galerkin Method in Space

We provide an error analysis of two methods for time stepping the wave equation. These are based on the Hybridizable Discontinuous Galerkin (HDG) method to discretize in space, and the continuous Galerkin method to discretize in time. Two variants of HDG are proposed: a dissipative method based on the standard numerical flux used for elliptic problems, and a non-dissipative method based on a new choice of the flux involving time derivatives. The analysis of the fully discrete problem is based on simplified arguments using projections rather than explicit interpolants used in previous work. Some numerical results are shown that illuminate the theory.

On the Stability of Continuous–Discontinuous Galerkin Methods for Advection–Diffusion–Reaction Problems

We consider a finite element method which couples the continuous Galerkin method away from internal and boundary layers with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. The stability properties of the coupled method are illustrated with a numerical experiment.

Hybridizable discontinuous Galerkin p‐adaptivity for wave propagation problems

SUMMARYA p‐adaptive hybridizable discontinuous Galerkin method for the solution of wave problems is presented in a challenging engineering problem. Moreover, its performance is compared with a high‐order continuous Galerkin. The hybridization technique allows to reduce the coupled degrees of freedom to only those on the mesh element boundaries, whereas the particular choice of the numerical fluxes opens the path to a superconvergent postprocessed solution. This superconvergent postprocessed solution is used to construct a simple and inexpensive error estimator. The error estimator is employed to obtain solutions with the prescribed accuracy in the area (or areas) of interest and also drives a proposed iterative mesh adaptation procedure. The proposed method is applied to a nonhomogeneous scattering problem in an unbounded domain. This is a challenging problem because, on the one hand, for high frequencies, numerical difficulties are an important issue because of the loss of the ellipticity and the oscillatory behavior of the solution. And on the other hand, it is applied to real harbor agitation problems. That is, the mild slope equation in frequency domain (Helmholtz equation with nonconstant coefficients) is solved on real geometries with the corresponding perfectly matched layer to damp the diffracted waves. The performance of the method is studied on two practical examples. The adaptive hybridizable discontinuous Galerkin method exhibits better efficiency compared with a high‐order continuous Galerkin method using static condensation of the interior nodes. Copyright © 2013 John Wiley & Sons, Ltd.

Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887---888, 2001; J Comput Phys 219:513---531, 2006; Comput Appl Math 218:79---87, 2008; J Comput Phys 228:2517---2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145---192, 2012; J Comput Appl Math 236:2317---2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915---4946, 2004; Int J Numer Methods Fluids 42:265---295, 2003; Kuzmin and Moller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525---558, 2002; J Comput Phys 198:131---158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming $$P_1$$ and $$Q_1$$ finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix---Raviart element (Crouzeix and Raviart in RAIRO R3 7:33---76, 1973) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher---Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97---111, 1992). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection---diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix---Raviart element. Good resolution of smooth and discontinuous profiles is attested to $$Q_1^\mathrm{nc}$$ approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix---vector multiplications is addressed.

Optimal Rate of Convergence for a Nonstandard Finite Difference Galerkin Method Applied to Wave Equation Problems

The optimal rate of convergence of the wave equation in both the energy and theL2-norms using continuous Galerkin method is well known. We exploit this technique and design a fully discrete scheme consisting of coupling the nonstandard finite difference method in the time and the continuous Galerkin method in the space variables. We show that, for sufficiently smooth solution, the maximal error in theL2-norm possesses the optimal rate of convergenceO(h2+(Δt)2)wherehis the mesh size andΔtis the time step size. Furthermore, we show that this scheme replicates the properties of the exact solution of the wave equation. Some numerical experiments should be performed to support our theoretical analysis.

Adjoint Based A Posteriori Analysis of Multiscale Mortar Discretizations with Multinumerics

In this paper we derive a posteriori error estimates for linear functionals of the solution to an elliptic problem discretized using a multiscale nonoverlapping domain decomposition method. The error estimates are based on the solution of an appropriately defined adjoint problem. We present a general framework that allows us to consider both primal and mixed formulations of the forward and adjoint problems within each subdomain. The primal subdomains are discretized using either an interior penalty discontinuous Galerkin method or a continuous Galerkin method with weakly imposed Dirichlet conditions. The mixed subdomains are discretized using Raviart--Thomas mixed finite elements. The a posteriori error estimate also accounts for the errors due to adjoint-inconsistent subdomain discretizations. The coupling between the subdomain discretizations is achieved via a mortar space. We show that the numerical discretization error can be broken down into subdomain and mortar components which may be used to drive adaptive refinement.