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In this paper, we introduce and analyze multilevel inexact Uzawa and Arrow–Hurwicz algorithms for solving saddle point problems. For the definition of the problem and that of the Uzawa and Arrow–Hurwicz algorithms, we adopt the framework introduced in I. Ekeland and R. Temam, convex analysis and variational problems, North-Holland Publishing Company, Amsterdam, Oxford, 1976, where the saddle point is defined by optimizations on convex sets. The results are obtained for Hilbert spaces, and therefore they can be applied to obtain convergence results for the multigrid methods in finite element spaces. We prove the convergence of the two algorithms, the multilevel Uzawa and the multilevel Arrow–Hurwicz algorithms. Also, we give new convergence proofs for the Uzawa and Arrow–Hurwicz algorithms themselves to better characterize their convergence.

This paper is to introduce a type of cascadic multigrid method for coupled semilinear elliptic equations. Instead of solving the coupled semilinear elliptic equation on a very fine finite element space directly, the new scheme needs to solve a decoupled linear system by some smoothing steps on each of multilevel finite element spaces and solve a coupled semilinear elliptic equation on a coarse space. By choosing the appropriate number of smoothing steps on different finite element spaces, the corresponding optimal convergence rate and optimal computation work can be derived. Besides, the adaptive cascadic multigrid method for coupled semilinear elliptic equations and its analysis are also presented theoretically and numerically. Moreover, the requirement of bounded second order derivatives of the nonlinear term in the existing multigrid methods is reduced to the Lipschitz continuation property in the presented new scheme. Some numerical experiments are presented to validate our theoretical analysis.

We present and analyze computationally Geometric MultiGrid (GMG) preconditioning techniques for Generalized Minimal RESidual (GMRES) iterations to space-time finite element methods (STFEMs) for a coupled hyperbolic–parabolic system modeling, for instance, flow in deformable porous media. By using a discontinuous temporal test basis, a time marching scheme is obtained. Higher order approximations that offer the potential to inherit most of the rich structure of solutions to the continuous problem on computationally feasible grids increase the block partitioning dimension of the algebraic systems, comprised of generalized saddle point blocks. Our V-cycle GMG preconditioner uses a local Vanka-type smoother. Its action is defined in an exact mathematical way. Due to nonlocal coupling mechanisms of 348 unknowns, the smoother is applied on patches of elements. This ensures damping of higher order error frequencies. By numerical experiments of increasing complexity, the efficiency of the solver for STFEMs of different polynomial order is illustrated and confirmed. Its parallel scalability is analyzed. Beyond this study of classical performance engineering, the solver’s energy efficiency is investigated as an additional and emerging dimension in the design and tuning of algorithms on the hardware.

In the present work, we address the coupling problem between fluid flow and porous media flow. A hybrid numerical method is proposed to solve the coupled Stokes-Darcy problem with Beavers-Joseph-Saffman (BJS) interface conditions. The method is presented in detail, rigorously tested, and its error estimates are thoroughly analyzed, demonstrating its robustness and effectiveness in handling the coupled system.

For solving the time fractional conservation laws with discontinuous solutions such as shock waves, current feasible methods are the implicit finite-volume TVD schemes that employ the Lax-Friedrichs fluxes corrected by the limited slopes. However, the schemes are hard to implement when the fractional order α is close to zero since in their implementation there are still no efficient methods to solve the strongly nonlinear spacial discrete systems at discrete times. We, aiming at increasing the implementation efficiency of the schemes above, develop two multigrid methods to solve these nonlinear spacial discrete systems. Numerical tests show that both multigrid methods have good convergence for relatively larger α , and the second one has an advantage in that its convergence is affected slightly when the order α decreases to zero.

Extrapolation cascadic multigrid (EXCMG) method is an efficient multigrid method which has mainly been used for solving the two-dimensional elliptic boundary value problems with linear finite element discretization in the existing literature. In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the L 2 -norm and L ∞ -norm for the solution and its gradient when the exact solution belongs to C 6 . Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.

A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fine finite element space, we transform it into a linear boundary value problem defined in each level of a multigrid sequence and a small-scale semilinear Neumann problem defined in a low-dimensional correction subspace. Furthermore, the linear boundary value problem can be efficiently solved using local and parallel methods. The proposed process derives an optimal error estimate with linear computational complexity. Additionally, compared with existing multigrid methods for semilinear Neumann problems that require bounded second order derivatives of nonlinear terms, ours only needs bounded first order derivatives. A rigorous theoretical analysis is proposed in this paper, which differs from the maturely developed theories for equations with Dirichlet boundary conditions.

In this work, we develop geometric multigrid algorithms for the immersed finite element methods for elliptic problems with interface (Chou et al. Adv. Comput. Math. 33 , 149–168 2010 ; Kwak and Lee, Int. J. Pure Appl. Math. 104 , 471–494 2015 ; Li et al. Numer. Math. 96 , 61–98 2003 , 2004 ; Lin et al. SIAM J. Numer. Anal. 53 , 1121–1144 2015 ). We need to design the transfer operators between levels carefully, since the residuals of finer grid problems do not satisfy the flux condition once projected onto coarser grids. Hence, we have to modify the projected residuals so that the flux conditions are satisfied. Similarly, the correction has to be modified after prolongation. Two algorithms are suggested: one for finite element spaces having vertex degrees of freedom and the other for edge average degrees of freedom. For the second case, we use the idea of conforming subspace correction used for P 1 nonconforming case (Lee 1993 ). Numerical experiments show the optimal scalability in terms of number of arithmetic operations, i.e., O ( N ) for V -cycle and CG algorithms preconditioned with V -cycle. In V -cycle, we used only one Gauss-Seidel smoothing. The CPU times are also reported.

A V-cycle multigrid method for the Hellan–Herrmann–Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity assumption. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context.

A multigrid multilevel Monte Carlo (MGMLMC) method is developed for the stochastic Stokes–Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Three interface conditions with randomness are considered on the interface between Stokes and Darcy equations, especially the Beavers–Joesph interface condition with random hydraulic conductivity. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes–Darcy model to improve the fidelity. Under suitable assumptions on the random coefficient, we prove the existence and uniqueness of the weak solution of the variational form. To construct the numerical method, we first adopt the Monte Carlo (MC) method and finite element method, for the discretization in the probability space and physical space, respectively. In order to improve the efficiency of the classical single-level Monte Carlo (SLMC) method, we adopt the multilevel Monte Carlo (MLMC) method to dramatically reduce the computational cost in the probability space. A strategy is developed to calculate the number of samples needed in MLMC method for the stochastic Stokes–Darcy model. In order to accomplish the strategy for MLMC method, we also present a practical method to determine the variance convergence rate for the stochastic Stokes–Darcy model with Beavers–Joseph interface condition. Furthermore, MLMC method naturally provides the hierarchical grids and sufficient information on these grids for multigrid (MG) method, which can in turn improve the efficiency of MLMC method. In order to fully make use of the dynamical interaction between this two methods, we propose a multigrid multilevel Monte Carlo (MGMLMC) method with finite element discretization for more efficiently solving the stochastic model, while additional attention is paid to the interface and the random Beavers–Joesph interface condition. The computational cost of the proposed MGMLMC method is rigorously analyzed and compared with the SLMC method. Numerical examples are provided to verify and illustrate the proposed method and the theoretical conclusions.