Two-grid mixed finite element method for nonlinear hyperbolic equations

A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the approximation of the pressure equation and mixed finite element method with characteristics for the concentration equation. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this full discrete scheme problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = H2 in this paper. Finally, numerical experiment indicates that two-grid algorithm is very effective.

Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

Two-grid algorithm for two-dimensional linear Schrödinger equation by the mixed finite element method

This paper studies and analyzes a model describing the flow of contaminated brines through the porous media under severe thermal conditions caused by the radioactive contaminants. The problem is approximated based on combining the mixed finite element method with the modified method of characteristics. In order to solve the resulting algebraic nonlinear equations efficiently, a two-grid method is presented and discussed in this paper. This approach includes a small nonlinear system on a coarse grid with size H and a linear system on a fine grid with size h . It follows from error estimates that asymptotically optimal accuracy can be obtained as long as the mesh sizes satisfy H = O h 1 / 3 .

A method for performing nested grid calculations with a large-eddy simulation code is described. A common numerical method is used for all meshes, and the grid architecture consists of a single outer or coarse grid, and nested or fine grids, which overlap in some common region. Inter-grid communication matches the velocity, pressure and potential temperature fields in the overlap region. Resolved and sub-grid scale (SGS) turbulent fluxes and kinetic energy on the fine grid are averaged to the coarse grid using a conservation rule equivalent to Germano's identity used to develop dynamic SGS models. Simulations of a slightly convective, strong shear planetary boundary layer were carried out with varying surface-layer resolutions. Grid refinements in the (x, y, z) directions of up to (5, 5, 2) times were employed. Two-way interaction solutions on the coarse and fine meshes are successfully matched in the overlap region on an instantaneous basis, and the turbulent motions on the fine grid blend smoothly into the coarse grid across the grid interface. With surface-layer grid nesting, significant increases in resolved eddy fluxes and variances are found. The energy-scale content of the vertical velocity, and hence vertical turbulent fluxes, appear to be most influenced by increased grid resolution. Vertical velocity spectra show that the dominant scale shifts towards higher wavenumbers (smaller scales) and the magnitude of the peak energy is increased by more than a factor of 3 with finer resolution. Outside of the nested region the average heat and momentum fluxes and spectra are slightly influenced by the fine resolution in the surface layer. From these results we conclude that fine resolution is required to resolve the details of the turbulent motions in the surface layer. At the same time, however, increased resolution in the surface layer does not appreciably alter the ensemble statistics of the resolved and SGS motions outside of the nested region.

This paper investigates L∞-estimates for the general optimal control problems governed by two-dimensional nonlinear elliptic equations with pointwise control constraints using mixed finite element methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. The authors derive L∞-estimates for the mixed finite element approximation of nonlinear optimal control problems. Finally, the numerical examples are given.

ABSTRACTA framework is proposed to bridge coarse and fine meshes in a single simulation within the phase field method for fracture. Fine meshes are used in the vicinity of localized defects to accurately capture crack initiation, while coarse meshes are used away from initial defects and include only crack propagation paths. This reduces the prohibitive computational times associated with uniformly fine meshes over the entire domain, or with the use of complex adaptive meshes in the phase field method. The coupling between the two overlapping meshes is achieved using a variational formulation in which the energies of the models associated with the fine and coarse meshes are weighted in the superposition zone. Two situations are considered. The first includes the resolution of the phase field problem only in the fine mesh, while the coarse mesh is limited to the undamaged elastic problem. In the second situation, cracks can propagate in both the fine and coarse mesh. Variational formulations and associated finite element implementations are detailed. Numerical examples are presented, showing the potential of this approach to significantly reduce computational costs in the phase field method for cracking without affecting the accuracy.

Efficient solver for mixed finite element method and control-volume mixed finite element method in 3-D on Hexahedral grids

In this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh sizeH, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh sizeor a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size. These methods we studied provide an approximate solutionwith the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh sizeh.Hence, our methods can save a large amount of computational time.

The influence of various aspects of Malaise trap design on efficiency in collecting various groups of Hymenoptera was compared using commercially available traps. The influence of fine and coarse mesh sizes and the importance of the incorporation of pan traps into the design were evaluated in two sampling periods in an old dune community at Pinery Provincial Park in southern Ontario. Numbers of individuals collected in Malaise trap heads and pan traps were presented for each paired comparison of mesh size (fine and coarse) and location (top and bottom of a hill) for various families of Hymenoptera. A graphical analysis of the normalized catch data for more inclusive groupings, viz. Symphyta, Aculeata, Ichneumonoidea, and microhymenoptera, was presented. The major results of these comparisons were as follows: (1) pan traps were an important component of efficient Malaise traps, especially for Aculeata and microhymenoptera; (2) coarse mesh was more effective in collecting Aculeata; (3) coarse and fine mesh were both effective in collecting Ichneumonoidea; and (4) fine mesh was more effective in collecting microhymenoptera. There was an interaction effect between the type of trap used and groups collected and it was not possible to maximize simultaneously the collection of all groups of Hymenoptera. The use of various mesh types and a trap design that incorporates pan traps was recommended.

Two-grid [formula omitted]–P1 mixed finite element methods combined with Crank–Nicolson scheme for a class of nonlinear parabolic equations

We present a locally mass conservative scheme for the approximation of two-phase flow in a porous medium that allows us to obtain detailed fine scale solutions on relatively coarse meshes. The permeability is assumed to be resolvable on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We define a two-scale mixed finite element space and resulting method, and describe in detail the solution algorithm. It involves a coarse scale operator coupled to a subgrid scale operator localized in space to each coarse grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid scale problems independently of the coarse grid approximation. The coarse grid problem is modified to take into account the subgrid scale solution and solved as a large linear system of equations posed over a coarse grid. Finally, the coarse scale solution is corrected on the subgrid scale, providing a fine grid representation of the solution. Numerical examples are presented, which show that near-well behavior and even extremely heterogeneous permeability barriers and streaks are upscaled well by the technique.

Two-level Picard and modified Picard methods for the Navier-Stokes equations