Galerkin Finite Element Discretization Research Articles

Extension of a second order velocity slip/temperature jump boundary condition to simulate high speed micro/nanoflows

In the current work, for the first time, we extend the application of a second order slip/jump equations introduced by Karniadakis et al. for the simulation of high speed, high Knudsen (Kn) number flows over a nano-scale flat plate and a micro-scale cylinder. The NS equations subject to a second order slip/jump boundary conditions are solved using the Petrov–Galerkin Finite Element discretization. We compare our numerical solution for flow and thermal field with the solution of the DSMC and Generalized Hydrodynamic (GH) techniques, as well as a recently developed slip/jump boundary condition, i.e., Paterson equation. Current results demonstrate the suitable accuracy of the employed boundary conditions for different set of test cases. Our numerical solutions are obtained with much less numerical costs compared to alternative boundary conditions.

Discontinuous Galerkin finite element discretization of a strongly anisotropic diffusion operator

SUMMARYThe discretization of a diffusion equation with a strong anisotropy by a discontinuous Galerkin finite element method is investigated. This diffusion term is implemented in the tracer equation of an ocean model, thanks to a symmetric tensor that is composed of diapycnal and isopycnal diffusions. The strong anisotropy comes from the difference of magnitude order between both diffusions. As the ocean model uses interior penalty terms to ensure numerical stability, a new penalty factor is required in order to correctly deal with the anisotropy of this diffusion. Two penalty factors from the literature are improved and established from the coercivity property. One of them takes into account the diffusion in the direction normal to the interface between the elements. After comparison, the latter is better because the spurious numerical diffusion is weaker than with the penalty factor proposed in the literature. It is computed with a transformed coordinate system in which the diffusivity tensor is diagonal, using its eigenvalue decomposition. Furthermore, this numerical scheme is validated with the method of manufactured solutions. It is finally applied to simulate the evolution of temperature and salinity due to turbulent processes in an idealized Arctic Ocean. Copyright © 2014 John Wiley & Sons, Ltd.

TRUNCATED HIERARCHICAL PRECONDITIONING FOR THE STOCHASTIC GALERKIN FEM

Stochastic Galerkin finite element discretizations of partial differential equations with coefficients characterized by arbitrary distributions lead, in general, to fully block dense linear systems. We propose two novel strategies for constructing preconditioners for these systems to be used with Krylov subspace iterative solvers. In particular, we present a variation on of the hierarchical Schur complement preconditioner, developed recently by the authors, and an adaptation of the symmetric block Gauss-Seidel method. Both preconditioners take advantage of the hierarchical structure of global stochastic Galerkin matrices, and also, when applicable, of the decay of the norms of the stiffness matrices obtained from the polynomial chaos expansion of the coefficients. This decay allows to truncate the matrix-vector multiplications in the action of the preconditioners. Also, throughout the global matrix hierarchy, we approximate solves with certain submatrices by the associated diagonal block solves. The preconditioners thus require only a limited number of stiffness matrices obtained from the polynomial chaos expansion of the coefficients, and a preconditioner for the diagonal blocks of the global matrix. The performance is illustrated by numerical experiments.

Optimal Distributed Control of Nonlocal Steady Diffusion Problems

A control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied. The control is the right-hand side forcing function and the objective of control is a standard matching functional. A recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system. When sufficient conditions on certain kernel functions and the volume constraints hold, the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived. We demonstrate the convergence, as the nonlocal interactions vanish, of the optimal nonlocal state to the optimal state of a local PDE-constrained control problem. We also define continuous and discontinuous Galerkin finite element discretizations of the optimality system for which we derive a priori error estimates. Numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained for the nonlocal diffusion equations and for PDEs. In particular, we observe that using nonlocal models result in a better match to nonsmooth target functions and a reduced cost of control. We also provide a one-dimensional numerical example of nonlocal heat conduction in a bar for which we determine the optimal heat source that results in an optimal match to a reference temperature distribution.

Indirect Multiple Shooting for Nonlinear Parabolic Optimal Control Problems with Control Constraints

We discuss the indirect multiple shooting approach for the solution of PDE-based parabolic optimal control problems with control constraints. The method is formulated within an abstract function space setting and uses a space-time Galerkin finite element discretization. The emphasis is on the embedding of indirect multiple shooting into the optimal control framework as well as the detailed description of the discretization within the PDE context. Numerical results for linear and nonlinear model problems with and without control constraints illustrate the efficient use of indirect multiple shooting particularly in cases where other standard methods fail.

Discontinuous Galerkin finite element discretization for steady Stokes flows with threshold slip boundary condition

This work is concerned with the discontinuous Galerkin finite approximations for the steady Stokes equations driven by slip boundary condition of “friction” type. Assuming that the flow region is a bounded, convex domain with a regular boundary, we formulate the problem and its discontinuous Galerkin approximations as mixed variational inequalities of the second kind with primitive variables. The well posedness of the formulated problems are established by means of a generalization of the Babuška-Brezzi theory for mixed problems. Finally, a priori error estimates using energy norm for both the velocity and pressure are obtained.

Convergence of linearized backward Euler–Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with temporal gauge

We present error analysis of fully discrete Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with the temporal gauge, where a linearized backward Euler scheme is used for the time discretization. We prove that the convergence rate is O(τ+hr) if the finite element space of piecewise polynomials of degree r is used. Due to the degeneracy of the problem, the convergence rate is one order lower than the optimal convergence rate of finite element methods for parabolic equations. Numerical examples are provided to support our theoretical analysis.

The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator

We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem. We demonstrate that, when sufficient conditions on certain kernel functions hold, the solution of the nonlocal equation converges to the solution of the fractional Laplacian equation on bounded domains as the nonlocal interactions become infinite. We also introduce a continuous Galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates. Through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite-volume constraints.

Output-based mesh adaptation for high order Navier–Stokes simulations on deformable domains

We present an output-based mesh adaptation strategy for Navier–Stokes simulations on deforming domains. The equations are solved with an arbitrary Lagrangian–Eulerian (ALE) approach, using a discontinuous Galerkin finite-element discretization in both space and time. Discrete unsteady adjoint solutions, derived for both the state and the geometric conservation law, provide output error estimates and drive adaptation of the space–time mesh. Spatial adaptation consists of dynamic order increment or decrement on a fixed tessellation of the domain, while a combination of coarsening and refinement is used to provide an efficient time step distribution. Results from compressible Navier–Stokes simulations in both two and three dimensions demonstrate the accuracy and efficiency of the proposed approach. In particular, the method is shown to outperform other common adaptation strategies, which, while sometimes adequate for static problems, struggle in the presence of mesh motion.

Block-Symmetric and Block-Lower-Triangular Preconditioners for PDE-Constrained Optimization Problems

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained optimization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices are derived. Numerical implementations show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.

Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media

Modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources. Seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geologic settings. Finite elements enable accurate simulation of time-dependent wave propagation in heterogeneous media. They are more costly than finite-difference methods, but this is compensated by their superior accuracy if the finite-element mesh follows the sharp impedance contrasts and by their improved efficiency if the element size scales with wavelength, hence with the local wave velocity. However, 3D complex geologic settings often contain details on a very small scale compared to the dominant wavelength, requiring the mesh to contain elements that are smaller than dictated by the wavelength. Also, limitations of the mesh generation software may produce regions where the elements are much smaller than desired. In both cases, this leads to a reduction of the time step required to solve the wave propagation and significantly increases the computational cost. Local time stepping (LTS) can improve the computational efficiency and speed up the simulation. We evaluated a local formulation of an LTS scheme with second-order accuracy for the discontinuous Galerkin finite-element discretization of the wave equation. We tested the benefits of the scheme by considering a geologic model for a North-Sea-type example.

A modified diffusion coefficient technique for the convection diffusion equation

A new modified diffusion coefficient (MDC) technique for solving convection diffusion equation is proposed. The Galerkin finite-element discretization process is applied on the modified equation rather than the original one. For a class of one-dimensional convection–diffusion equations, we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size, then, prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients. The application of the derived analytic formula of MDC is extended for other classes of convection–diffusion equations, where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction. The numerical results of the proposed approach for two-dimensional, variable coefficients, with boundary layers, convection-dominated problems show stable and accurate solutions even on coarse grids. Accordingly, multigrid based solvers retain their efficient convergence rates.

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

For a class of block two-by-two systems of linear equations with certain skew-Hamiltonian coefficient matrices, we construct additive block diagonal preconditioning matrices and discuss the eigen-properties of the corresponding preconditioned matrices. The additive block diagonal preconditioners can be employed to accelerate the convergence rates of Krylov subspace iteration methods such as MINRES and GMRES. Numerical experiments show that MINRES preconditioned by the exact and the inexact additive block diagonal preconditioners are effective, robust and scalable solvers for the block two-by-two linear systems arising from the Galerkin finite-element discretizations of a class of distributed control problems.

Galerkin finite element methods for semilinear elliptic differential inclusions

In this paper weconsider Galerkin finite element discretizations of semilinear elliptic differentialinclusions that satisfy a relaxed one-sided Lipschitz condition.The properties of the set-valued Nemytskii operators are discussed, andit is shown that the solution sets of both, the continuous and the discretesystem, are nonempty, closed, bounded, and connected sets in $H^1$-norm.Moreover, the solution sets of the Galerkin inclusionconverge with respect to the Hausdorff distance measured in $L^p$-spaces.

High-order explicit local time-stepping methods for damped wave equations

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams–Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

High-order computational methods for option valuation under multifactor models

Many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second-order discretization schemes. A higher-order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high-order convergence rates for the solution of multi-factor option problems. These new schemes employ Galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two-asset option problems and time integration of the resulting semi-discrete systems requires the computation of a single matrix exponential. The computations indicate that this combination of high-order finite elements and exponential time integration leads to efficient algorithms for multi-factor problems. Highly accurate European prices are obtained with relatively coarse meshes and high-order convergence rates are also observed for options with the American early exercise feature. Various numerical examples are provided for illustrating the accuracy of the option prices for Heston’s and Bates stochastic volatility models and for two-asset problems under Merton’s jump-diffusion model.

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge–Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge–Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space–time discontinuous Galerkin finite element discretization of the advection–diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained.

A supercoarsening multigrid method for poroelasticity in 3D coupled flow and geomechanics modeling

The Galerkin finite-element discretization of the force balance equation typically leads to large linear systems for geomechanical problems with realistic dimensions. In iteratively coupled flow and geomechanics modeling, a large linear system is solved at every timestep often multiple times during coupling iterations. The iterative solution of the linear system stemming from the poroelasticity equations constitutes the most time-consuming and memory-intensive component of coupled modeling. Block Jacobi, LSOR, and Incomplete LU factorization are popular preconditioning techniques used for accelerating the iterative solution of the poroelasticity linear systems. However, the need for more effective, efficient, and robust iterative solution techniques still remains especially for large coupled modeling problems requiring the solution of the poroelasticity system for a large number of timesteps. We developed a supercoarsening multigrid method (SCMG) which can be multiplicatively combined with commonly used preconditioning techniques. SCMG has been tested on a variety of coupled flow and geomechanics problems involving single-phase depletion and multiphase displacement of in-situ hydrocarbons, CO2 injection, and extreme material property contrasts. Our analysis indicates that the SCMG consistently improves the convergence properties of the linear systems arising from the poroelasticity equations, and thus, accelerates the coupled simulations for all cases subject to investigation. The joint utilization of the two-level SCMG with the ILU1 preconditioner emerges as the most optimal preconditioning/iterative solution strategy in a great majority of the problems evaluated in this work. The BiCGSTAB iterative solver converges more rapidly compared to PCG in a number of test cases, in which various SCMG-accelerated preconditioning strategies are applied to both iterators.

Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid methods.

Discontinuous Galerkin methods for problems with Dirac delta source

In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L^2-norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement framework. Numerical examples for the symmetric interior penalty scheme are presented which confirm the theoretical results.