On a multigrid solver for the three-dimensional Biot poroelasticity system in multilayered domains

In this paper, we present a multigrid method with problem-dependent prolongation and restriction for the three-dimensional Biot poroelasticity system in a multilayered domain. The system is discretized on a staggered grid using the finite volume method. In the discretization special care is taken of the coefficients’ discontinuity. The prolongation and restriction operators are derived in a consistent manner with the discretization, so that they account for the discontinuities of the coefficients, as well as for the coupling of the unknowns within the Biot system. A set of numerical experiments shows necessity of use of operator-dependent restriction and prolongation in the multigrid solver for the considered class of problems. This research has been partially supported by the Kaiserslautern Excellence Cluster Dependable Adaptive Systems and Mathematical Modelling, the Spanish project MEC/FEDER MTM 2004-019051, the Diputación General de Aragón, and by the INTAS project 03-50-4395.     

Multigrid method for nonlinear poroelasticity equations

In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies.     

An efficient multigrid solver for a reformulated version of the poroelasticity system

In this paper, we present a robust and efficient multigrid solver for a transformed version of the system of poroelasticity equations. The transformation enables us to treat the system in a decoupled fashion. We show that the transformation boils down to a stabilization term in the iterative scheme, and that the solution of the original problem is identical to the solution of the transformed problem. A highly efficient multigrid method can be developed, confirmed by numerical experiments.     

An Efficient Multigrid Solver based on Distributive Smoothing for Poroelasticity Equations

In this paper, we present a robust distributive smoother in a multigrid method for the system of poroelasticity equations. Within the distributive framework, we deal with a decoupled system, that can be smoothed with basic iterative methods like an equation-wise red-black Jacobi point relaxation. The properties of the distributive relaxation are optimized with the help of Fourier smoothing analysis. A highly efficient multigrid method results, as is confirmed by Fourier two-grid analysis and numerical experiments.AMS Subject Classifications: 65N55, 74F10, 74S10, 65M12.This revised version was published online in December 2004. In the previous version the name of the first author was wrong.     

Ordering Techniques for Two- and Three-Dimensional Convection-Dominated Elliptic Boundary Value Problems

Multigrid methods with simple smoothers have been proven to be very successful for elliptic problems with no or only moderate convection. In the presence of dominant convection or anisotropies as it might appear in equations of computational fluid dynamics (e.g. in the Navier-Stokes equations), the convergence rate typically decreases. This is due to a weakened smoothing property as well as to problems in the coarse grid correction. In order to obtain a multigrid method that is robust for convection-dominated problems, we construct efficient smoothers that obtain their favorable properties through an appropriate ordering of the unknowns. We propose several ordering techniques that work on the graph associated with the (convective part of the) stiffness matrix. The ordering algorithms provide a numbering together with a block structure which can be used for block iterative methods. We provide numerical results for the Stokes equations with a convective term illustrating the improved convergence properties of the multigrid algorithm when applied with an appropriate ordering of the unknowns. Received July 12, 1999; revised October 1, 1999     

Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods

The advent of parallel computers has led to the development of new solution algorithms for time-dependent partial differential equations. Two recently developed methods, multigrid waveform relaxation and time-parallel multigrid, have been designed to solve parabolic partial differential equations on many time-levels simultaneously. This paper compares the convergence properties of these methods, based on the results of an exponential Fourier mode analysis for a model problem.     

A mixed strategy of combining evolutionary algorithms with multigrid methods

Multigrid methods have been proven to be an efficient approach in accelerating the convergence rate of numerical algorithms for solving partial differential equations. This paper investigates whether multigrid methods are helpful to accelerate the convergence rate of evolutionary algorithms for solving global optimization problems. A novel multigrid evolutionary algorithm is proposed and its convergence is proven. The algorithm is tested on a set of 13 well-known benchmark functions. Experiment results demonstrate that multigrid methods can accelerate the convergence rate of evolutionary algorithms and improve their performance.     

Mode shapes and frequencies by finite element method using consistent and lumped masses

The rate of convergence of the mode shapes and frequencies by the finite element method using consistent and lumped mass formulations has been established. Simple examples are given to demonstrate the results. It has been found that for a system of differential equations of second order such as the equations of equilibrium in terms of displacement in the theory of elasticity, membrane etc., a proper lumped mass formulation will not suffer any loss of rate of convergence utilizing simple elements. However, in the case of higher order differential equations or when the use of more complicated elements is required or desired, a consistent mass formulation often will provide a better rate of convergence.     

A Robust Multigrid Algorithm for the Simulation of a Yawed Flat Plate

This paper presents a full multigrid solver for the simulation of a flow over a yawed flat plate. The two problems associated with this simulation; boundary layers and entering flows with non-aligned characteristics, have been successfully overcome through the combination of a plane-implicit solver and semicoarsening. In fact, this multigrid algorithm exhibits a textbook multigrid convergence rate, i.e., the solution of the discrete system of equations is obtained in a fixed amount of computational work, independently of the grid size, grid stretching factor and non-alignment parameter. Also, a parallel variant of the smoother based on a four-color ordering of planes is investigated.     

A multigrid solver for two-dimensional stochastic diffusion equations

Steady and unsteady diffusion equations, with stochastic diffusivity coefficient and forcing term, are modeled in two dimensions by means of stochastic spectral representations. Problem data and solution variables are expanded using the Polynomial Chaos system. The approach leads to a set of coupled problems for the stochastic modes. Spatial finite-difference discretization of these coupled problems results in a large system of equations, whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time. To accelerate the convergence of the iterative technique, a multigrid method, based on spatial coarsening, is implemented. Numerical experiments show good scaling properties of the method, both with respect to the number of spatial grid points and the stochastic resolution level.     

An efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

Semicoarsening smoothers for the simulation of a flat plate at yaw

This paper presents a full multigrid solver for the simulation of flow over a yawed flat plate. The two problems associated with this simulation; boundary layers and entering flows with non-aligned characteristics, have been successfully overcome through the combination of a plane-implicit solver and semicoarsening. In fact, this multigrid algorithm exhibits a textbook multigrid convergence rate, i.e., the solution of the discrete system of equations is obtained in a fixed amount of computational work, independently of the grid size, grid stretching factor and non-alignment parameter. Also, a parallel variant of the smoother based on a four-color ordering of planes is investigated.     

Numerical analysis of confined turbulent flow

This work outlines a second order accurate, coupled, conservative new numerical scheme for solving a two dimensional incompressible turbulent flow filed. Mean vorticity, ω, and mean stream function, ψ, are used as the mean flow dependent variables. The turbulent kinetic energy $\text{k}$ and the turbulent energy decay rate, ?, are used to define the turbulent state. In the present computational scheme two systems of equations and variables are considered: the mean flow system, ψ-ω, and the turbulent state system, $\text{k-?}$. Every system is solved implicity in a coupled double loop manner, and all the flow equations are solved iteratively in the global sense. Since the turbulence boundary conditions have a non-regular variation near a solid wall, they are coupled to the equations implicitly in both systems. In this way the numerical instabilities due to the irregular form of the equations near the solid walls are suppressed. The rate of convergence of the new numerical scheme of the coupled systems ψ-ω and $\text{k-?}$ is twice that realized when solving these equations separately. The necessary conditions for convergence of the numerical equations are investigated as well as the rate of convergence features. The detailed stability conditions are derived. As an example, the axisymmetric mixing of two confined jets with an internal heat source is considered with this numerical scheme.     

Parallel computing as a vehicle for engineering design of complex functional surfaces

Thin liquid film flow over surfaces containing complex multiply connected topography is modelled using lubrication theory. The resulting time dependent nonlinear coupled set of governing equations for film thickness and pressure is solved on different parallel computing platforms using a purpose written portable and scalable parallel multigrid algorithm in order to achieve the fine-scale resolution required to guarantee mesh independent solutions. The robustness of the approach is demonstrated via the solution of three problems: one to establish the convergence characteristics viz. the partitioning and message passing strategies adopted, taking flow over a well-defined trench topography as a benchmark against existing experimental and corresponding numerical predictions; two, flow through a sparsely distributed set of occlusions with computations performed on different parallel architectures; three, free-surface planarisation with respect to flow over complex topography - the first an engineered functional substrate, the second a naturally occurring surface.     

Convergence analysis of geometrical multigrid methods for solving data-sparse boundary element equations

The convergence analysis of multigrid methods for boundary element equations arising from negative-order pseudo-differential operators is quite different from the usual finite element multigrid analysis for elliptic partial differential equations. In this paper, we study the convergence of geometrical multigrid methods for solving large-scale, data-sparse boundary element equations. In particular, we investigate multigrid methods for \(\mathcal{H}\)-matrices arising from the adaptive cross approximation to the single layer potential operator.     

An algorithm for ideal multigrid convergence for the steady Euler equations

A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Flows in two-dimensional channels and over airfoils have been computed. Using Gauss–Seidel relaxation with the grid vertices ordered in the flow direction, ideal multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier–Stokes equations.     

Nested multigrid methods for time-periodic,parabolic optimal control problems

We present a nested multigrid method to optimize time-periodic, parabolic, partial differential equations (PDE). We consider a quadratic tracking objective with a linear parabolic PDE constraint. The first order optimality conditions, given by a coupled system of boundary value problems can be rewritten as an Fredholm integral equation of the second kind, which is solved by a multigrid of the second kind. The evaluation of the integral operator consists of solving sequentially a boundary value problem for respectively the state and the adjoints. Both problems are solved efficiently by a time-periodic space-time multigrid method.     

A Multigrid Method of the Second Kind for Solving Linear Systems of Odes Discretized by Continuous Runge-Kutta Methods

A Waveform Relaxation method as applied to a linear system of ODEs is the Picard iteration for a linear Volterra integral equation of the second kind ({\cal I} - {\cal K})y = b \eqno (1) called Waveform Relaxation second kind equation. A corresponding Waveform Relaxation Runge-Kutta method is the Picard iteration for a discretized version ({\cal I} - {\cal K}_l )y_l = b_l \eqno (2) of the integral equation (1), where y l is the continuous solution of the original linear system of ODE provided by the so called limit method. We consider a W-cycle multigrid method, with Picard iteration as smoothing step, for iteratively computing y l . This multigrid method belongs to the class of multigrid methods of the second kind as described in Hackbusch [3, chapter 16]. In the paper we prove that the truncation error after one iteration is of the same order of the discretization error y l @ y of the limit method and the truncation error after two iterations has order larger than the discretization error. Thus we can see the multigrid method as a new numerical method for solving the original linear system of ODE which provides, after one iteration, a continuous solution of the same order of the solution of the limit method, and after two iterations, a solution with asymptotically the same error of the solution of the limit method. On the other hand the computational cost of the multigrid method is considerably smaller than the limit method.     

A multigrid solver for phase field simulation of microstructure evolution

This paper presents a semi-implicit numerical method for the simulation of grain growth in two dimensions with a multi-phase field model. To avoid the strong stability condition of traditional explicit methods, a first-order, semi-implicit discretisation scheme is employed, which offers a good compromise with regard to memory intensity and computational requirements. A nonlinear multigrid solver based on the Full Approximation Scheme is implemented to solve the equations resulting from this discretisation. Simulations with the multigrid solver show that the solver has grid size independent convergence properties and is faster than a standard first-order explicit solver. As such, the multigrid solver promises to be a reliable additional computational tool for the simulation of microstructural evolution. A comparison with existing alternatives remains, however, subject of further investigation. To validate the implementation, the results of specific test cases are studied.     

Development of a multigrid code for 3-D Navier-Stokes equations and its application to a grid-refinement study

A multigrid acceleration technique has been developed to solve the three-dimensional Navier-Stokes equations efficiently. An explicit multistage Runge-Kutta type-of time-stepping scheme is used as the basic algorithm in conjunction with the multigrid scheme. A grid-refinement study has been conducted to obtain grid converged solutions for transonic flow over a finite wing. Present solutions indicate that the number of multigrid cycles required to achieve a given level of convergence does not increase with the number of mesh points employed, making it a very attractive scheme for fine meshes.