A Semi-Uniform Multigrid Algorithm for Solving Elliptic Interface Problems

We introduce a new geometric multigrid algorithm to solve elliptic interface problems. First we discretize the problems by the usual {P_{1}}-conforming finite element methods on a semi-uniform grid which is obtained by refining a uniform grid. To solve the algebraic system, we adopt subspace correction methods for which we use uniform grids as the auxiliary spaces. To enhance the efficiency of the algorithms, we define a new transfer operator between a uniform grid and a semi-uniform grid so that the transferred functions satisfy the flux continuity along the interface. In the auxiliary space, the system is solved by the usual multigrid algorithm with a similarly modified prolongation operator. We show {\mathcal{W}}-cycle convergence for the proposed multigrid algorithm. We demonstrate the performance of our multigrid algorithm for problems having various ratios of parameters. We observe that the computational complexity of our algorithms are robust for all problems we tested.

Cascadic Newton’s method for the elliptic Monge–Ampère equation

In this paper, a cascadic Newton’s method is designed to solve the Monge–Ampère equation. In the process of implementing the cascadic multigrid, we use the Full-Local type interpolation as prolongation operator and Newton iteration as smoother. In order to obtain Full-Local type interpolation, we provide several finite difference stencils. Especially, the skewed finite difference methods are first applied by us for the elliptic Monge–Ampère equation. Based on Full-Local interpolation techniques and cascade principle, the new algorithm can save a large amount of computation time. Some numerical experiments are provided to confirm the efficiency of our proposed method.

Use of Multiple Multiscale Operators To Accelerate Simulation of Complex Geomodels

Summary Multiscale methods have been developed as a robust alternative to upscaling and to accelerate reservoir simulation. In their basic setup, multiscale methods use both a restriction operator to construct a reduced system of flow equations that can be solved on a coarser grid and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. When combined with a local smoother, this gives an iterative solver that can efficiently compute approximate pressures to within a prescribed accuracy and still provide mass-conservative fluxes. We present an adaptive and flexible framework for combining multiple sets of such multiscale approximations. Each multiscale approximation can target a certain scale; geological features such as faults, fractures, facies, or other geobodies; or a particular computational challenge such as propagating displacement and chemical fronts, wells being turned on or off, and others. Multiscale methods that fit the framework are characterized by three features. First, the prolongation and restriction operators are constructed by use of a nonoverlapping partition of the fine grid. Second, the prolongation operator is composed of a set of basis functions, each of which has compact support within a support region that contains a coarse gridblock. Finally, the basis functions form a partition of unity. These assumptions are quite general and encompass almost all existing multiscale (finite-volume) methods that rely on localized basis functions. The novelty of our framework is that it enables multiple pairs of prolongation and restriction operators—computed on different coarse grids and possibly also by different basis-function formulations—to be combined into one iterative procedure. Through a series of numerical examples consisting of both idealized geology and flow physics as well as a geological model of a real asset, we demonstrate that the new iterative framework increases the accuracy and efficiency of the multiscale technology by improving the rate at which one converges the fine-scale residuals toward machine precision. In particular, we demonstrate how it is possible to combine multiscale prolongation operators that have different spatial resolution and that each individual operator can be designed to target, among others, challenging grids, including faults, pinchouts, and inactive cells; high-contrast fluvial sands; fractured carbonate reservoirs; and complex wells.

Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

In this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.

Aerodynamic shape design using hybrid evolutionary computing and multigrid-aided finite-difference evaluation of flow sensitivities

Purpose – The purpose of this paper is to propose an accurate and efficient technique for computing flow sensitivities by finite differences of perturbed flow fields. It relies on computing the perturbed flows on coarser grid levels only: to achieve the same fine-grid accuracy, the approximate value of the relative local truncation error between coarser and finest grids unperturbed flow fields, provided by a standard multigrid method, is added to the coarse grid equations. The gradient computation is introduced in a hybrid genetic algorithm (HGA) that takes advantage of the presented method to accelerate the gradient-based search. An application to a classical transonic airfoil design is reported. Design/methodology/approach – Genetic optimization algorithm hybridized with classical gradient-based search techniques; usage of fast and accurate gradient computation technique. Findings – The new variant of the prolongation operator with weighting terms based on the volume of grid cells improves the accuracy of the MAFD method for turbulent viscous flows. The hybrid GA is capable to efficiently handle and compensate for the error that, although very limited, is present in the multigrid-aided finite-difference (MAFD) gradient evaluation method. Research limitations/implications – The proposed new variants of HGA, while outperforming the simple genetic algorithm, still require tuning and validation to further improve performance. Practical implications – Significant speedup of CFD-based optimization loops. Originality/value – Introduction of new multigrid prolongation operator that improves the accuracy of MAFD method for turbulent viscous flows. First application of MAFD evaluation of flow sensitivities within a hybrid optimization framework.

Multigrid multiblock hovering rotor solutions

The effect of multigrid acceleration implemented within an upwind-biased Euler method for hovering rotor flows is presented. Previous work has considered multigrid convergence for structured single block rotor solutions. However, for forward flight simulation a multiblock approach is essential and, hence, the flow-solver has been extended to include multigrid acceleration within a multiblock solver. The requirement to capture the vortical wake development over several turns means a long numerical integration time is required for hovering rotors, and the solution (wake) away from the blade is significant. Hence, the solution evolution and convergence is different to a fixed wing case where convergence depends primarily on propagating errors away from the surface as quickly as possible, and multigrid acceleration is shown here to be less effective for hovering rotor flows. Previous single block simulations demonstrated that a simple multigridV-cycle was the most effective, smoothing in the decreasing mesh density direction only, with a relaxed trilinear prolongation operator. This is also shown to be the case for multiblock simulations. Results are presented for multigrid computations with 2, 3, and 4, mesh levels, and a CPU reduction of approximately 80% is demonstrated for 4 mesh levels.

Multigrid acceleration of an upwind Euler method for hovering rotor flows

The effect of multigrid acceleration implemented within an upwind-biased Euler method for hovering rotor flows is presented. The requirement to capture the vortical wake development over several turns means a long numerical integration time is required for hovering rotors, and the solution (wake) away from the blade is significant. Furthermore, the flow in the region near the blade root is effectively incompressible. Hence, the solution evolution and convergence is different to a fixed wing case where convergence depends primarily on propagating errors away from the surface as quickly as possible, and multigrid acceleration is shown to be less effective for hovering rotor flows. It is found that a simple V-cycle is the most effective, smoothing in the decreasing mesh density direction only, with a relaxed trilinear prolongation operator. Results are presented for multigrid computations with 2, 3, 4, and 5 mesh levels, and a CPU reduction of approximately 80% is demonstrated for five mesh levels.