Coarse Grid Solution Research Articles

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

This article presents and studies a two-level grad-div stabilized finite element discretization method for solving numerically the steady incompressible Navier–Stokes equations. The method consists of two steps. In the first step, we compute a rough solution by solving a nonlinear Navier–Stokes system on a coarse grid. And then, in the second step, we pass the coarse grid solution to a fine grid to linearize the nonlinear term, update the solution by solving a linearized problem based on Newton iterations. In both steps, a grad-div stabilization term is incorporated into the system to reduce the influence of pressure on the approximate velocity. We analyze stability and asymptotic convergence of the approximate solutions, derive explicit dependence of the solution errors on the grad-div stabilization parameter and viscosity. We perform also some numerical tests to validate the theoretical analysis and illustrate the efficiency of the proposed method. Compared with the standard two-level method without stabilizations, the grad-div stabilization term added in present method improves the accuracy of the approximate velocity, accelerates the convergence of the nonlinear iterations for the coarse mesh nonlinear system, and reduces the computational time.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations

Abstract In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and L 2 L^{2} -norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.

Shock capturing using discontinuous Galerkin method and overset grids for two-dimensional Euler equations

A new procedure to capture shocks which confines the shock capturing algorithms to overset grids is proposed and demonstrated for steady state flows containing straight and bow shocks using a discontinuous Galerkin solver for two dimensional Euler equations. Preliminary results for capturing moving straight shocks are also presented. Troubled cell data, obtained using a coarse grid, is used to determine the approximate location of the shock. A refined and structured overset grid aligned to the shock is constructed based on this information. The solver is run again with the coarse grid solution as the initial condition using a shock capturing overset grid to give a solution with the shock aligned to the grid line. Solutions obtained with a coarse grid alone, overset grid aligned to the shock, and one not aligned to the shock are used to demonstrate the efficacy of the scheme for problems having analytical solutions. Results for supersonic flow over a ramp, shock reflecting off a flat plate, the supersonic flow over a circular cylinder, and a simple Mach reflection are presented. For the flow over a circular cylinder, the results obtained are validated using an existing analytical method for calculating the shock offset distance. Since there is a manual component in the construction of overset grids, as a first step towards eliminating it, a moving normal shock is also captured accurately using an overset grid at the location of the shock with an arbitrary Lagrangian-Eulerian discontinuous Galerkin method.

Numerical extensions to incorporate subgrid corrections in an established storm surge model

ABSTRACT Inundation models represent coastal regions with a grid of computational points, often with varying resolution of flow pathways and barriers. Models based on coarse grid solutions of shallow water equations have been improved recently via the use of subgrid corrections, which account for information (ground surface elevations, roughness characteristics) at smaller scales. In this work, numerical approaches of an established storm surge model are extended to include subgrid corrections. In an attempt to maintain continuity with existing users and results, model extensions were limited to those needed to provide basic subgrid capabilities, and included two major additions. First, a finite volume method is used to incorporate corrections to the mass and momentum equations using high-resolution ground surface elevations. Second, the no-slip condition imposed on the B-grid wet/dry interface in the model is modified to a slip condition to enable flows in channels with widths comparable to cell size. Numerical results demonstrate these numerical extensions can significantly enhance the accuracy of the model’s predictions of coastal flooding, with low additional computational cost.

Computational Modeling for High-Fidelity Coarsening of Shallow Water Equations Based on Subgrid Data

Small-scale features of shallow water flow obtained from direct numerical simulation (DNS) with two different computational codes for the shallow water equations are gathered offline and subsequently employed with the aim of constructing a reduced-order correction. This is used to facilitate high-fidelity online flow predictions at much reduced costs on coarse meshes. The resolved small-scale features at high resolution represent subgrid properties for the coarse representation. Measurements of the subgrid dynamics are obtained as the difference between the evolution of a coarse grid solution and the corresponding DNS result. The measurements are sensitive to the particular numerical methods used for the simulation on coarse computational grids and can be used to approximately correct the associated discretization errors. The subgrid features are decomposed into empirical orthogonal functions (EOFs), after which a corresponding correction term is constructed. By increasing the number of EOFs in the approximation of the measured values the correction term can in principle be made arbitrarily accurate. Both computational methods investigated here show a significant decrease in the simulation error already when applying the correction based on the dominant EOFs only. The error reduction accounts for the particular discretization errors that incur and are hence specific to the particular simulation method that is adopted. This improvement is also observed for very coarse grids which may be used for computational model reduction in geophysical and turbulent flow problems.

Two‐grid stabilized algorithms for the steady Navier–Stokes equations with damping

This paper presents and studies three two‐grid stabilized quadratic equal‐order finite element algorithms based on two local Gauss integrations for the steady Navier–Stokes equations with damping. In these algorithms, we first solve a stabilized nonlinear problem on a coarse grid, and then pass the coarse grid solution to a fine grid and solve a stabilized linear problem. Using some nonlinear analysis techniques, we analyze stability of the algorithms and derive optimal order error estimates of the approximate solutions. Theoretical and numerical results show that, when the algorithmic parameters are chosen appropriately, the accuracy of the approximate solutions computed by our two‐grid stabilized algorithms is comparable to that of solving a fully stabilized nonlinear problem on the same fine grid; however, our two‐grid algorithms save a large amount of CPU time than the one‐grid stabilized algorithm.

Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media

In this paper, we consider the poroelasticity problem in fractured and heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements in heterogeneous media. Due to scale disparity, many approaches have been developed for solving detailed fine-grid problems on a coarse grid. However, some approaches can lack good accuracy on a coarse grid and some corrections for coarse-grid solutions are needed. In this paper, we present a coarse-grid approximation based on the generalized multiscale finite element method (GMsFEM). We present the construction of the offline and online multiscale basis functions. The offline multiscale basis functions are precomputed for the given heterogeneity and fracture network geometry, where for the construction, we solve a local spectral problem and use the dominant eigenvectors (appropriately defined) to construct multiscale basis functions. To construct the online basis functions, we use current information about the local residual and solve coupled poroelasticity problems in local domains. The online basis functions are used to enrich the offline multiscale space and rapidly reduce the error using residual information. Only with appropriate offline coarse-grid spaces can one guarantee a fast convergence of online methods. We present numerical results for poroelasticity problems in fractured and heterogeneous media. We investigate the influence of the number of offline and online basis functions on the relative errors between the multiscale solution and the reference (fine-scale) solution.

Two-grid method for compressible miscible displacement problem by mixed finite element methods and expanded mixed finite element method of characteristics

A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium by employing a mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics (CEMFEM) for the concentration equation. The key point is to use a two-grid scheme to linearize the nonlinear term in the coupling equations. The main procedure of the algorithm is to solve small scaled nonlinear equations on the coarse grid and to deal with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Then, it is shown that a two-grid algorithm achieves optimal approximation as long as the mesh sizes satisfy $H = O(h^{\frac {1}{2}})$ . Finally, numerical experiments confirmed the numerical analysis of two-grid algorithm.

Multiscale simulations for multi-continuum Richards equations

In this paper, we study a multiscale method for simulating a dual-continuum unsaturated flow problem within complex heterogeneous fractured porous media. Mathematically, each of the dual continua is modeled by a multiscale Richards equation (for pressure head), and these equations are coupled to one another by transfer terms. On its own, Richards equation is already a nonlinear partial differential equation, and it is exceedingly difficult to solve numerically due to the extra nonlinear dependencies involving the soil water. To deal with multiple scales, our strategy is that starting from a microscopic scale, we upscale the coupled system of dual-continuum Richards equations via homogenization by the two-scale asymptotic expansion, to obtain a homogenized system, at an intermediate scale (level). Based on a hierarchical approach, the homogenization’s effective coefficients are computed through solving the arising cell problems. To tackle the nonlinearity, after time discretization, we use Picard iteration procedure for linearization of the homogenized Richards equations. At each Picard iteration, some degree of multiscale still remains from the intermediate level, so we utilize the generalized multiscale finite element method (GMsFEM) combining with a multi-continuum approach, to upscale the homogenized system to a macroscopic (coarse-grid) level. This scheme involves building uncoupled and coupled multiscale basis functions, which are used not only to construct coarse-grid solution approximation with high accuracy but also (with the coupled multiscale basis) to capture the interactions among continua. These prospects and convergence are demonstrated by several numerical results for the proposed method.

Two-grid finite element method for the dual-permeability-Stokes fluid flow model

In this paper, two-grid finite element method for the steady dual-permeability-Stokes fluid flow model is proposed and analyzed. Dual-permeability-Stokes interface system has vast applications in many areas such as hydrocarbon recovery process, especially in hydraulically fractured tight/shale oil/gas reservoirs. Two-grid method is popular and convenient to solve a large multiphysics interface system by decoupling the coupled problem into several subproblems. Herein, the two-grid approach is used to reduce the coding task substantially, which provides computational flexibility without losing the approximate accuracy. Firstly, we solve a global problem through standard Pk − Pk− 1 − Pk − Pk finite elements on the coarse grid. After that, a coarse grid solution is applied for the decoupling between the interface terms and the mass exchange terms to solve three independent subproblems on the fine grid. The three independent parallel subproblems are the Stokes equations, the microfracture equations, and the matrix equations, respectively. Four numerical tests are presented to validate the numerical methods and illustrate the features of the dual-permeability-Stokes model.

A method for preserving large-scale flow patterns in low-resolution ocean simulations

It is typical for low-resolution simulations of the ocean to miss not only small- but also large-scale patterns of the flow dynamics compared with their high-resolution analogues. It is usually attributed to the inability of coarse-grid models to properly reproduce the effects of the unresolved small-scale dynamics on the resolved large scales. In part, the reason for that is that coarse-grid models fail to at least keep the coarse-grid solution within the region of phase space occupied by the true solution (the high-resolution solution projected onto the coarse grid). In this paper we offer a solution to this problem by computing the image point in the phase space restricted to the region of the true flow dynamics. The proposed method shows encouraging results for both low- and high-dimensional phase spaces, it takes a near-zero effort to implement into existing numerical codes and has ample room for further improvements.

Multiscale Model Reduction of the Unsaturated Flow Problem in Heterogeneous Porous Media with Rough Surface Topography

In this paper, we consider unsaturated filtration in heterogeneous porous media with rough surface topography. The surface topography plays an important role in determining the flow process and includes multiscale features. The mathematical model is based on the Richards’ equation with three different types of boundary conditions on the surface: Dirichlet, Neumann, and Robin boundary conditions. For coarse-grid discretization, the Generalized Multiscale Finite Element Method (GMsFEM) is used. Multiscale basis functions that incorporate small scale heterogeneities into the basis functions are constructed. To treat rough boundaries, we construct additional basis functions to take into account the influence of boundary conditions on rough surfaces. We present numerical results for two-dimensional and three-dimensional model problems. To verify the obtained results, we calculate relative errors between the multiscale and reference (fine-grid) solutions for different numbers of multiscale basis functions. We obtain a good agreement between fine-grid and coarse-grid solutions.

Neural networks for multicontinuum models

Neural networks are developed for multicontinuum models. Multicontinuum models [1] assume that processes divided into several continuums and they are connected by some exchange member. In problems connected with PDE not only solution, but also coefficients can depends on time. First we construct LSTM neural network for exchange coefficient that depend on time. We compare the results obtained on the basis of the reference coefficient and on the basis of the neural network. Next we apply same neural network on the solutions. The proposed approach involves the use of a neural network for the approximate generation of a solutions and the further adaptation of the solution of a neural network for generate coarse grid solution based on DG finite elements.

A salinity cut-off method to control numerical dispersion in low-salinity waterflooding simulation

Low-salinity or controlled salinity waterflooding (LSWF) is a promising enhanced oil recovery (EOR) technique. In simulations of this process, numerical dispersion smears saturation fronts, causing errors in the results. The objective of this work is to control these effects in LSWF simulation. We examine the impact of numerical dispersion on simulated LSWF performance. The low-salinity (LS) front is smeared even at unfeasibly fine grids. The velocities of the water fronts are altered. A numerical mixed zone forms around the interface between the injected and resident brines. This mimics a typical physical mixing effect.In reservoir simulation, threshold salinities are defined where the low salinity effect (LSE) is first encountered. It has been suggested that numerical dispersion effects can be corrected by imposing effective thresholds. We demonstrate that existing methods to evaluate these effective salinities do not accurately predict the salt front movement especially when dispersion is significant.We propose a simple simulation-based approach to evaluate the effective salinities based on the conservation of volumes of the resident and injected brines in the reference and upscaled solutions. After comparing analytical and corrected coarse-grid solutions in one-dimension, the effectiveness of the approach is demonstrated in multi-dimensional systems.A method is proposed to control the numerical mixed zone to replicate a physical longitudinal mixing effect. This method is demonstrated in one-dimension and does not require a fine-grid numerical solution as a benchmark. We investigate the effects of effective thresholds on the modeled transverse mixing or dispersion. A method to model transverse dispersion in simulations with an effective longitudinal component is suggested. This method is extended to the explicit modeling of physical dispersion in systems with transverse flows.We can now simply evaluate the effective salinities for a simulation grid; and control its numerical dispersion effects to representative physical mixing or dispersion.

Reduced-order deep learning for flow dynamics. The interplay between deep learning and model reduction

In this paper, we investigate neural networks applied to multiscale simulations of porous media flows and discuss a design of a novel deep neural network model reduction approach for multiscale problems. Due to the multiscale nature of the medium, the fine-grid resolution gives rise to a huge number of degrees of freedom. In practice, low-order models are derived to reduce the computational cost. In our paper, we use a non-local multicontinuum (NLMC) approach [18], which represents the solution on a coarse grid for porous media flows. In specific, we construct a reduced dimensional space which takes into account multi-continuum information. The numerical solutions are then sought in such space. Using multi-layer learning techniques, we formulate and learn input-output maps constructed with NLMC on a coarse grid. We study the features of the coarse-grid solutions that neural networks capture via relating the input-output optimization to ℓ1 minimization of flow solutions. In proposed multi-layer networks, we can learn the forward operators in a reduced way without computing them as in POD like approaches. We present soft thresholding operators as activation function, which our studies show to have some advantages. With these activation functions, the neural network identifies and selects important multiscale features which are crucial in modeling the underlying flow. Using trained neural network approximation of the input-output map, we construct a reduced-order model for the solution approximation. We use multi-layer networks for the time stepping and reduced-order modeling, where at each time step the appropriate important modes are selected. For a class of nonlinear flow problems, we suggest an efficient strategy. Numerical examples are presented to examine the performance of our method.

Numerical solution of two-dimensional nonlinear Schrödinger equation using a new two-grid finite element method

A new two-grid finite element scheme is presented for two-dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the nonlinear system using the coarse-grid solution as the initial guess, and furthermore one more linear system on the coarse space is solved. The error estimations of the two-grid solution in the L2 and H1 norm are given. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and two-grid algorithm still achieves optimal approximation as long as the mesh sizes satisfy H=O(h13).

A two-level fully discrete finite element variational multiscale method for the unsteady Navier–Stokes equations

A two-level fully discrete finite element variational multiscale method based on two local Gauss integrations for the unsteady Navier–Stokes equations is presented and studied, where conforming finite element pairs are used for the spatial discretization and a three-point difference formula is employed for the temporal discretization. At each time step of this method, a stabilized nonlinear Navier–Stokes system is first solved on a coarse grid, and then a stabilized linear problem is solved on a fine grid to correct the coarse grid solution, where the numerical forms of the Navier–Stokes equations both on coarse and fine meshes are stabilized by a stabilization term defined by the difference between a consistent and an under-integrated matrix of the gradient of velocity interpolant. Error bound of the approximate velocity is derived. Numerical results are also given to verify the theoretical predictions and demonstrate the effectiveness of the proposed numerical method.

Computation of volume potentials on structured grids with the method of local corrections

We present a new version of the Method of Local Corrections (MLC) \cite{mlc}, a multilevel, low communications, non-iterative, domain decomposition algorithm for the numerical solution of the free space Poisson's equation in 3D on locally-structured grids. In this method, the field is computed as a linear superposition of local fields induced by charges on rectangular patches of size $O(1)$ mesh points, with the global coupling represented by a coarse grid solution using a right-hand side computed from the local solutions. In the present method, the local convolutions are further decomposed into a short-range contribution computed by convolution with the discrete Green's function for an $Q^{th}$-order accurate finite difference approximation to the Laplacian with the full right-hand side on the patch, combined with a longer-range component that is the field induced by the terms up to order $P-1$ of the Legendre expansion of the charge over the patch. This leads to a method with a solution error that has an asymptotic bound of $O(h^P) + O(h^Q) + O(\epsilon h^2) + O(\epsilon)$, where $h$ is the mesh spacing, and $\epsilon$ is the max norm of the charge times a rapidly-decaying function of the radius of the support of the local solutions scaled by $h$. Thus we have eliminated the low-order accuracy of the original method (which corresponds to $P=1$ in the present method) for smooth solutions, while keeping the computational cost per patch nearly the same with that of the original method. Specifically, in addition to the local solves of the original method we only have to compute and communicate the expansion coefficients of local expansions (that is, for instance, 20 scalars per patch for $P=4$). Several numerical examples are presented to illustrate the new method and demonstrate its convergence properties.

Convergence Properties of Local Defect Correction Algorithm for the Boundary Element Method

Sometimes boundary value problems have isolated regions where the solution changes rapidly. Therefore, when solving numerically, one needs a fine grid to capture the high activity. The fine grid can be implemented as a composite coarse-fine grid or as a global fine grid. One cheaper way of obtaining the composite grid solution is the use of the local defect correction technique. The technique is an algorithm that combines a global coarse grid solution and a local fine grid solution in an iterative way to estimate the solution on the corresponding composite grid. The algorithm is relatively new and its convergence properties have not been studied for the boundary element method. In this paper the objective is to determine convergence properties of the algorithm for the boundary element method. First, we formulate the algorithm as a fixed point iterative scheme, which has also not been done before for the boundary element method, and then study the properties of the iteration matrix. Results show that we can always expect convergence. Therefore, the algorithm opens up a real alternative for application in the boundary element method for problems with localised regions of high activity.