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This paper proposes and investigates the two-grid stabilized lowest equal-order finite element method for the time-independent dual-permeability-Stokes model with the Beavers-Joseph-Saffman-Jones interface conditions. This method is mainly based on the idea of combining the two-grid and the two local Gauss integrals for the dual-permeability-Stokes system. In this technique, we use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the dual-permeability-Stokes model using the lowest equal-order finite element quadruples. In the two-grid scheme, the global problem is solved using the standard P 1 - P 1 - P 1 - P 1 finite element approximations only on a coarse grid with grid size H. Then, a coarse grid solution is applied on a fine grid of size h to decouple the interface terms and the mass exchange terms for solving the three independent subproblems such as the Stokes equations, microfracture equations, and the matrix equations on the fine grid. On the other hand, microfracture and matrix equations are decoupled through the mass exchange terms. The weak formulation is reported, and the optimal error estimate is derived for the two-grid schemes. Furthermore, the numerical results validate that the two-grid stabilized lowest equal-order finite element method is effective and has the same accuracy as the coupling scheme when we choose h = H 2 . [ABSTRACT FROM AUTHOR]

In this paper, we propose a two-grid algorithm for solving parabolic equation with nonlinear compressibility coefficient, spatially discretized by the weak Galerkin finite element method. The optimal error estimates are established. We further show that both grid solutions can achieve the same accuracy as long as the grid size satisfies H = O (h 1 / 2). Compared with Newton iteration, the two-grid algorithm could greatly reduce the computational cost. We verify the effectiveness of the algorithm by performing numerical experiments. [ABSTRACT FROM AUTHOR]

In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size HH and the fine grid size hh satisfy h=O(H(2k+1)⁄(k+1))h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) (k≥1k\ge 1), where kk is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

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 퐻, 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 -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. [ABSTRACT FROM AUTHOR]

In this paper, a two-grid characteristic block-centered finite difference method is developed for solving a nonlinear Darcy-Forchheimer slightly compressible miscible displacement problem in porous media. The method is proposed for solving the nonlinear system in two steps. In the first step, the nonlinear equations are approximated on a coarse grid using the characteristic block-centered finite difference method. In the second step, the nonlinear system is linearized using Newton's method with a small positive parameter to ensure the differentiability of the nonlinear term | u | u in the Darcy-Forchheimer equation. The proposed two-grid method is rigorously analyzed, in which a priori error estimates of the velocity, pressure, concentration and its flux are provided, and the rates of convergence are O (ε + Δ t + H 3 + h 2). Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods by comparing the efficiency, especially in terms of CPU time. [ABSTRACT FROM AUTHOR]

Based on local Gauss integral technique and backtracking technique, this paper presents and studies three kinds of two-grid stabilized finite element algorithms for the stationary Navier-Stokes equations. The proposed methods consist of deducing a coarse solution on the nonlinear system, updating the solution on a fine mesh via three different methods, and solving a linear correction problem on the coarse mesh to obtain the final solution. The error estimates are derived for the solution approximated by the proposed algorithms. A series of numerical experiments are illustrated to test the applicability and efficiency of our proposed methods, and support the theoretical analysis results. [ABSTRACT FROM AUTHOR]

In this paper, the Marker and Cell scheme based on a two-grid algorithm is proposed for the two-dimensional incompressible Darcy–Brinkman–Forchheimer equations in porous media. The motivation of the two-grid Marker and Cell algorithm is figuring out a nonlinear equation on a coarse grid with mesh size H and a linear equation on a fine grid with mesh size h. A small positive parameter ε is introduced. By using it, the non-differentiable nonlinear term can be transformed into the term which is twice continuously differentiable. The error estimates of the velocity and pressure in the L 2 norms are obtained, which show O (ε + H 4 + h 2) . Second-order accuracy for some terms of velocity in the H 1 norms is also obtained. Several numerical experiments are provided to confirm the availability of this efficient second-order algorithm. Behavior of the fluid flow with different Brinkman number is considered. [ABSTRACT FROM AUTHOR]

Based on two-grid discretizations and quadratic equal-order finite elements for the velocity and pressure approximations, we develop a three-step defect-correction stabilized algorithm for the incompressible Navier-Stokes equations, where non-homogeneous Dirichlet boundary conditions are considered and high Reynolds numbers are allowed. In this developed algorithm, we first solve an artificial viscosity stabilized nonlinear problem on a coarse grid in a defect step and then correct the resulting residual by solving two stabilized and linearized problems on a fine grid in correction steps. While the fine grid correction problems have the same stiffness matrices with only different right-hand sides. We use a variational multiscale method to stabilize the system, making the algorithm has a broad range of potential applications in the simulation of high Reynolds number flows. Under the weak uniqueness condition, we give a stability analysis of the present algorithm, analyze the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we perform a series of numerical examples to demonstrate the promise of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The semi‐resolved Computational Fluid Dynamics coupled with the Discrete Element Method (CFD‐DEM) method has emerged as approach to modeling particle‐fluid interactions in granular materials with high particle size ratios. However, challenges arise from conflicting requirements regarding the CFD grid size, which must adequately resolve fluid flow in the pore space while maintaining a physically meaningful porosity field. This study addresses these challenges by introducing a two‐grid mapping approach. Initially, the porosity field associated with fine particles is estimated using a coarse CFD grid, which is then mapped to a dynamically refined grid. To ensure conservation of total solid volume, a volume compensation procedure is implemented. The proposed method has been rigorously verified using benchmark cases, showing its high computational efficiency and accurate handling of complex porosity calculations near the surface of coarse particles. Moreover, the previously unreported impact of the empirical drag correlation on fluid‐particle force calculations for both coarse and fine particles has been revealed. [ABSTRACT FROM AUTHOR]

In the paper, we consider the penalty finite element methods (FEMs) for the stationary Smagorinsky model. Firstly, a one‐grid penalty FEM is proposed and analyzed. Since this method is nonlinear, a novel linearized iteration scheme is derived for solving it. We also derived the stability and convergence of numerical solutions for this iteration scheme. Furthermore, a two‐grid penalty FEM is developed for Smagorinsky model. Under ε<

In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in L2$$ {L}^2 $$‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in L2$$ {L}^2 $$‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method. [ABSTRACT FROM AUTHOR]

We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L 2 -norm with order O (τ + h k + 1) . Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L 2 -norm with order O (τ + h k + 1 + H k + 2) . The numerical experiment is provided to demonstrate the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a new two-grid partially penalized immersed finite element method for solving the semilinear parabolic interface problems with meshes independent of the coefficient discontinuity. Based on the corresponding time-discrete system, we can unconditionally derive the optimal error estimates in both the L 2 norm and semi- H 1 norm, while previous works always require the coupling condition of time step and space size (e.g. condition τ = O (H)). Then, we design a two-grid algorithm based on Newton iteration to deal with nonlinear source term. It is shown, both theoretically and numerically, that the algorithm can achieve asymptotically optimal approximation in L 2 norm (or semi- H 1 norm) when the mesh size satisfies H = O (h 3 / 2) (or H = O (h 3)). [ABSTRACT FROM AUTHOR]

A two‐grid finite element scheme is presented for nonlinear Schrödinger equation. H1$$ {H}^1 $$ superconvergence error estimates of finite element solutions based on L∞$$ {L}^{\infty } $$ bound of finite element solutions is proved. The optimal order error estimations of the two‐grid solution in the Lp$$ {L}^p $$ and H1$$ {H}^1 $$ norm are given without any time‐step size conditions. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse, with no loss in the order of accuracy. [ABSTRACT FROM AUTHOR]

This paper develops a fully discrete two-grid mixed finite element scheme of a phase field model for the two immiscible and incompressible flows. In the model, the Navier-Stokes equations and Allen-Cahn equation are coupled by a phase function in which an extra phase induced stress term is added in the Navier-Stokes equations and a fluid induced transport term is added in the Allen-Cahn equation. We use the two-grid method to overcome the computational difficulties arising from the strong nonlinear coupling system. The stability and convergence of the two-grid method are obtained under the proper assumption of time discretization size k , space discretization size h and interface thickness ε. The analysis shows that the two-grid methods can get the same convergence as the classical one-grid methods provided that we choose proper coarse and fine mesh sizes. However, the two-grid methods can save much computational time compared with the one-grid methods. Some numerical experiments are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

A modified-upwind with block-centred finite difference scheme on the basis of the two-grid algorithm is presented for the convection-diffusion-reaction equations. This scheme can keep second-order accuracy in spatial mesh sizes for both state variables and fluxes in the convection–diffusion–reaction problem. Moreover, the two-grid algorithm is constructed in order to solve semilinear convection-dominated problems efficiently, in which the main idea is to settle an original semilinear equation on the coarse space, and next to settle a linearized equation on the fine space. The error estimate of the method proposed in this paper is given through theoretical analysis. It is indicated that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O (h 1 / 2). Thus, solving such a large-scale nonlinear problem is as easy as linearized problems. Besides, there are some numerical experiments to corroborate in practice that the algorithm is effective and robust to solve convection–diffusion–reaction problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-grid finite element method (FEM) for two-dimensional nonlinear unsaturated soil water ow problems. The application of two-grid methods can transform the solution of nonlinear problems on the fine grid to the solution of nonlinear problems on the coarse grid and the solution of linear problems on the fine grid. Based on the error results of standard FEM, we derive the continuous and discrete time error estimates in the energy norm for the two-grid method. Numerical experiments are carried out with the linear element as an example to illustrate that the two-grid method can achieve the similar optimal accuracy as standard FEM with less time as long as the two mesh sizes satisfy H = O(√h). [ABSTRACT FROM AUTHOR]

In this article, a weak Galerkin (WG) method is first presented and analyzed for the quasi‐linear elliptic problem of non‐monotone type. By using Brouwer's fixed point technique, the existence of WG solution and error estimates in both the energy norm and the L2$$ {L}^2 $$ norm are derived. Then an efficient two‐grid WG method is introduced to improve the computational efficiency. The convergence error of the two‐grid WG method is analyzed in the energy norm. Numerical experiments are presented to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Based on spatial finite-element methods combined with classical L1 time stepping method, the superconvergence analysis of the two-grid approximate scheme for the two-dimensional nonlinear time-fractional diffusion equations is considered. First, we use the rectangular Lagrange type finite element of order p to get a two-grid fully discrete scheme of the equation and discuss the superclose error estimate with the order O h p + 1 + H 2 p + 2 + τ 2 - α in the H 1 norm, here τ , H and h denote time step, coarse and fine grid sizes, respectively. Second, through the interpolated postprocessing approach, the global superconvergence of order O h 2 + H 4 + τ 2 - α in the H 1 norm is obtained. Finally, two numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, by combining the two-grid method with a partition of unity, a local and parallel partition of unity scheme is designed and investigated for the mixed Navier-Stokes-Darcy problem. The main features of the present method are the following: (1) Once a coarse solution is derived, both the linearized Navier-Stokes and Darcy subproblems with a finer grid are independent, which allows a parallel computing with little communication; (2) a partition of unity is considered to assemble the solutions obtained from the local subdomains to arrive at a global continuous approximation; (3) a further coarse grid correction is carried out to derive optimal error bounds for the fluid velocity and piezometric head in L2-norm. Moreover, the convergence of the proposed method is shown. Some numerical experiments are reported to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the unconditional convergence and error estimates of a two-grid finite element method for the semilinear parabolic integro-differential equations. By using a temporal-spatial error splitting technique, optimal L p and H 1 error estimates of the finite element method can be obtained. Moreover, to deal with the semilinearity of the model, we use the two-grid method. Optimal error estimates in L 2 and H 1 -norms of two-grid solution are derived without any time-step size conditions. Finally, some numerical results are provided to confirm the theoretical analysis, and show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We present in this paper a fully discrete two-grid finite element method to solve the diffusive Peterlin viscoelastic model on the basis of studying the fully discrete standard finite element method. The fully discrete two-grid method involves solving small nonlinear coarse spatial grid systems and linear fine spatial grid systems with the same time step. Under the reasonable assumptions, it is proven theoretically that the accuracy of fully discrete two-grid method is the same as the fully discrete finite element method on the fine spatial grid. Numerical examples are given to validate the accuracy and convergence order of the standard finite element method and two-grid method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze the two-grid immersed finite element methods for semi-linear parabolic interface problems with discontinuous diffusion coefficients. The immersed finite element methods are used for spatial discretization where the meshes are not aligned with the interface. Optimal error estimates have been derived for both spatially semi-discrete schemes and fully discrete schemes. The two-grid algorithms based on the Newton methods are adopted to treat the nonlinear term. It is theoretically and numerically illustrated that the two-grid immersed finite element methods can achieve optimal convergence order when the coarse mesh satisfies H = O (h 1 / 2) (or H = O (h 1 / 4)). [ABSTRACT FROM AUTHOR]

In this paper, we construct a backward Euler full-discrete two-grid finite volume element scheme for solving time-dependent Schrödinger equation. Combining the idea of the two-grid discretization, the original coupling system is solved in the coarse grid space, and the decoupling system with two independent Poisson problems is solved in the fine grid space, which ensures the accuracy and improves the computational efficiency. We further prove the optimal error estimate of the scheme rigorously. Finally, the numerical simulation results show that the two-grid method is more effective than the standard finite volume element method in solving coupled partial differential problems. [ABSTRACT FROM AUTHOR]

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 1 2 ) . Finally, numerical experiments confirmed the numerical analysis of two-grid algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a Crank-Nicolson block-centered finite difference method is first developed and analyzed for the nonlinear regularized long wave equation. By using a cutoff technique, second-order convergences both in time and space are proved under a suitable time-space stepsize constraint condition. To further improve the computational efficiency, an efficient two-grid block-centered finite difference method is introduced and analyzed, in which a resulting small-scale nonlinear problem is first solved on a coarse grid space of size H , and then a resulting large-scale linear problem is solved on a fine grid space of size h. Under a rough time-space stepsize constraint condition Δ t = o (H 1 / 4) , optimal-order error estimates for both the primal variable and its flux are derived on non-uniform spatial grids. Thus, the proposed method is competitive both in accuracy and efficiency compared with the fully nonlinear Crank-Nicolson block-centered finite difference scheme. Numerical experiments are presented to verify the theoretical analysis. • An efficient TG-CN-BCFD method is proposed for the RLW equation. • Optimal-order error estimates on non-uniform spatial grids are rigorously proved. • The estimate is derived under a very rough constraint condition Δ t = o (H 1 / 4). • Numerical experiments are given to show accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, based on two-grid discretizations and superposition principle, a new local and parallel finite element method is presented and studied for the coupled Stokes–Darcy model. Superposition principle is used to generate a series of local and independent subproblems to obtain a global continuous approximation. Optimal error estimates are derived. Numerical results are reported to support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this article, some local and parallel finite element methods are proposed and investigated for the time‐dependent convection–diffusion problem. With backward Euler scheme for the temporal discretization, the basic idea of the present methods is that for a solution to the considered equations, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure at each time step. The partition of unity is used to collect the local high frequency components to assemble a global continuous approximation. Theoretical results are obtained and numerical tests are reported to support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, an optimal error analysis of a three step two‐grid method for the equations of motion arising in the 2D Oldroyd model of order one is discussed. The model, which can be thought of as an integral perturbation of Navier‐Stokes equations (NSE), represents linear viscoelastic fluid flows. This non‐linear model is analyzed here using a three step numerical scheme. In the first step the problem is solved on a coarse grid, and we use this course grid solution to linearize the problem and solve it in second step, on a finer grid. Third step is a correcting step done on the finer grid. Optimal error estimates for the velocity in L∞(L2) and L∞(H1)‐norms and for the pressure in L∞(L2)‐norm in the semidiscrete case are established. Estimates are shown to be uniform under the uniqueness assumption. Then, based on backward Euler method, a completely discrete scheme is analyzed and optimal a priori error estimates are derived. All the analysis are carried out for the non‐smooth initial data. Finally we present some numerical results to validate our theoretical results. These examples show that the three step two‐grid method is efficient than solving a nonlinear problem directly, as is expected. [ABSTRACT FROM AUTHOR]

In this paper, we present a family of two-grid algorithms for semi-linear parabolic interface problems based on Partially penalized immersed finite element discretizations. Optimal a priori error estimates are derived both in the energy norm and L 2 norm, under the standard piecewise H 2 regularity assumption for the exact solution. For the nonlinear right hand side, we investigate two-grid methods base on Newton method. The efficiency of the two-grid methods is confirmed theoretically and numerically. [ABSTRACT FROM AUTHOR]

A two-grid method is proposed to solve phase field models, arising from modeling the band gap optimization of photonic crystal design based on the Transverse Electric field (TE mode). In our approach, finite element methods on nested grids are used to solve different Partial Differential Equations (PDEs), say, the Hemholtz equation for the state constrain and the phase field equation for topology evolving, respectively. Dicretization on the coarse and fine grids are performed to balance the costs between the phase evolving and state equation solver. To fuse the two scheme together, additional correction step is necessary in two-grid settings. Furthermore, convergence analysis for the proposed two-grid scheme is established. Finally, numerical examples are presented to show the efficiency and robustness, and it is illustrated that the two-grid method improves the speed of computation while remain the computational accuracy. [ABSTRACT FROM AUTHOR]

Based on two-grid discretizations, local and parallel finite element algorithms are proposed and analyzed for the time-dependent Oseen equations. Using conforming finite element pairs for the spatial discretization and backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Oseen equations using a coarse grid on the entire domain, and then correct the resulted residual using a fine grid on overlapped subdomains by some local and parallel procedures at each time step. By the theoretical tool of local a priori estimate for the fully discrete finite element solution, error bounds of the approximate solutions from the algorithms are estimated. Numerical results are also given to demonstrate the efficiency of the algorithms. [ABSTRACT FROM AUTHOR]

We present a numerical model for the approximation of multiphase flows with free surfaces and strong interfacial effects. The model relies on the multiphase incompressible Navier-Stokes equations, and includes surface tension effects on the interfaces between phases, and contact angles. The volume-of-fluid approach is used to track the interfaces and the free surfaces between the various phases and the ambient air. The numerical method relies on an operator splitting strategy. The space discretization relies on a two-grid approach that uses an unstructured finite element mesh for diffusion phenomena and a structured Cartesian grid for advection phenomena. An adaptive mesh refinement algorithm is incorporated to better track the interfaces and free surfaces on the finite element mesh, and approximate more accurately surface forces. The model is validated through numerical experiments, in particular for emulsion problems. • A numerical method for 3D incompressible Newtonian multiphase flows. • A discretization based on an operator splitting algorithm and a two-grid method. • A mesh cutting adaptive algorithm for the accurate approximation of interfacial effects. • The numerical approximation of surface tension and contact angles. • The numerical simulation of emulsion processes. [ABSTRACT FROM AUTHOR]

In this paper, based on two-grid discretization, a new local and parallel finite element method is proposed and investigated for the mixed Navier–Stokes–Darcy problem. Compared with the classical local and parallel finite element method, the main features of our method are: (1) partition of unity method is considered to generate local subproblems; (2) global continuous approximations are achieved. We theoretically analyze the proposed method and report on some numerical tests to support the theoretical results. [ABSTRACT FROM AUTHOR]

Two new two-grid algorithms are proposed for solving the Maxwell eigenvalue problem. The new methods are based on the two-grid methodology recently proposed by Xu and Zhou [Math. Comp., 70 (2001), pp. 17-25] and further developed by Hu and Cheng [Math. Comp., 80 (2011), pp. 1287-1301] for elliptic eigenvalue problems. The new two-grid schemes reduce the solution of the Maxwell eigenvalue problem on a fine grid to one linear indefinite Maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid. The new schemes, therefore, save total computational cost. The error estimates reveals that the two-grid methods maintain asymptotically optimal accuracy, and the numerical experiments presented confirm the theoretical results.

A combined method consisting of the mixed finite element method for the pressure equation and finite element method with characteristics for the concentration equation is proposed to solve the coupled system of incompressible two‐phase flow in porous media. Two‐grid algorithm based on the Newton iteration method is developed and analyzed for the nonlinear coupled system. It is shown, both theoretically and numerically, that our schemes are efficient and achieve asymptotically optimal accuracy as long as the mesh sizes satisfy H=O(h1/2). [ABSTRACT FROM AUTHOR]

Abstract The main aim of this paper is to apply a low order nonconforming EQ1rot $\mathit{EQ}_{1}^{\mathrm{rot}}$ finite element to solve the nonlinear Schrödinger equation. Firstly, the superclose property in the broken H1 $H^{1}$-norm for a backward Euler fully-discrete scheme is studied, and the global superconvergence results are deduced with the help of the special characters of this element and the interpolation postprocessing technique. Secondly, in order to reduce computing cost, a two-grid method is developed and the corresponding superconvergence error estimates are obtained. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations. The electric potential equation is approximated by a mixed finite element method, and the concentration equations are approximated by a standard Galerkin method. We estimate the error of the numerical solutions in the sense of the Lq norm. To linearize the full discrete scheme of the problem, we present an efficient two-grid method based on the idea of Newton iteration. The main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine grid. Error estimation for the two-grid solutions is analyzed in detail. It is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H = O(h1/2). Numerical experiments are given to illustrate the efficiency of the two-grid method. [ABSTRACT FROM AUTHOR]

In this paper, two-grid algorithms based on expanded mixed finite element method are presented for solving two-dimensional semilinear time fractional reaction-diffusion equations. To obtain the fully discrete scheme, the classical L 1 scheme is considered in the time direction, and the expanded mixed finite element method is used to approximate spatial direction. Then the error estimates and stability of fully discrete scheme are derived. To linearize the nonlinear system, the two-grid method based on Newton iteration are constructed. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment is presented further confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Based on two-grid discretizations, local and parallel finite element algorithms are studied for the Stokes equations with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Thereafter, the variational inequality can be transform into the variational identity as a multiplier in a convex set. The main idea of our algorithms is to approximate the low frequencies of the finite element solution using a coarse grid and use a fine grid to correct the resulted residual (that includes mostly high frequencies of the solution) by some local and parallel procedures. Error bounds for the approximate solutions are estimated. Numerical results are also given to demonstrate the effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel two-grid method of Eulerian–Lagrangian localized adjoint method (ELLAM)-mixed finite element method (MFEM) solutions for nonlinear and coupled miscible displacement problems, which describe complex fluid flow processes in porous media. An ELLAM is used to solve the nonlinear convection–diffusion equation for concentration, and we present a MFEM to compute the pressure equation for the pressure and Darcy velocity. A two-grid scheme is investigated to linearize and decouple the mixed-method equations. It is illustrated that the coarse space can be selected as coarse as H = O (h 1 ∕ 2) and the asymptotically optimal approximation as the nonlinear schemes can be still achieved. As a result, a large class of such nonlinear and coupled problems can be solved efficiently with large time steps. [ABSTRACT FROM AUTHOR]

We design and analyze a new finite element discretization technique based on iterative two-grid methods for semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that just applies the classical two-grid for semilinear elliptic PDEs in a successive fashion, namely, to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space. Thus, we only need to deal with semilinear problems in the coarse space in the first algorithm. Secondly, we replace the semilinear term by using the corresponding first order Taylor expansion in the second iterative two-grid. We provide the error estimates for the second iterative algorithm and present numerical experiments to show the efficiency of our methods. [ABSTRACT FROM AUTHOR]

Summary: A combined method consisting of mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics(CEMFEM) for the concentration equation is presented to solve the coupled system of incompressible miscible displacement problem. To solve the resulting nonlinear system of equations efficiently, the two‐grid algorithm relegates all of the Newton‐like iterations to grids much coarser than the original one, with no loss in order of accuracy. It is shown that coarse space can be extremely coarse and our algorithm achieve asymptotically optimal approximation when the mesh sizes satisfy H=O(h14). Numerical experiment is provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present two efficient two-grid algorithms for solving two-dimensional semi-linear elliptic interface problems using finite element method. To linearize the finite element equations, the Newton iteration approach and correction technique are applied. The new two-grid schemes reduce the solution of the semi-linear interface problem on a fine grid to one linear interface equation on the same fine grid and an original interface problem on a much coarser grid. Therefore, the new schemes save total computational cost. Theoretical analysis shows that the two-grid methods maintain asymptotically optimal accuracy, and the numerical experiments presented confirm the theoretical results. [ABSTRACT FROM AUTHOR]