Multigrid Preconditioner for Nonconforming Discretization of Elliptic Problems with Jump Coefficients

In this paper, we present a multigrid V-cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix–Raviart discretization of second-order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rates of the (multiplicative) two-grid and multigrid V-cycle algorithms will deteriorate rapidly because of large jumps in coefficient. However, the preconditioned systems have only a fixed number of small eigenvalues depending on the large jump in coefficient, and the effective condition numbers are independent of the coefficient and bounded logarithmically with respect to the mesh size. As a result, the two-grid or multigrid preconditioned conjugate gradient algorithm converges nearly uniformly. We also comment on some major differences of the convergence theory between the nonconforming case and the standard conforming case. Numerical experiments support the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.

Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted\(L^{2}\) -projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.

Scaling function and wavelet preconditioners for second order elliptic problems

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.MSC:35J30, 35J40.

The original additive average Schwarz method and its variants were introduced for solving second order elliptic boundary value problems. In this paper, we extend the original idea to fourth order problems by designing a variant of the additive average Schwarz method for the plate bending problem using the nonconforming Morley finite element for the discretization. Like the original average method this method uses only nonoverlapping subdomains, and its coarse space does not need any explicit coarse mesh. An analysis of the method, and some numerical experiments to validate the theory are presented.

In this work, we construct energy-minimizing coarse spaces for the finite element discretization of mixed boundary value problems for displacements in compressible linear elasticity. Motivated from the multiscale analysis of highly heterogeneous composite materials, basis functions on a triangular coarse mesh are constructed, obeying a minimal energy property subject to global pointwise constraints. These constraints allow that the coarse space exactly contains the rigid body translations, while rigid body rotations are preserved approximately. The application is twofold. Resolving the heterogeneities on the finest scale, we utilize the energy-minimizing coarse space for the construction of robust two-level overlapping domain decomposition preconditioners. Thereby, we do not assume that coefficient jumps are resolved by the coarse grid, nor do we impose assumptions on the alignment of material jumps and the coarse triangulation. Weonly assume that the size of the inclusions is small compared to the coarse mesh diameter. Ournumerical tests show uniform convergence rates independent of the contrast in the Young’s modulus within the heterogeneous material. Furthermore, we numerically observe the properties of the energy-minimizing coarse space in an upscaling framework. Therefore, we present numerical results showing the approximation errors of the energy-minimizing coarse space w.r.t. the fine-scale solution.

Second order elliptic problems with discontinuous coefficients are considered. The problem is discretized by the finite element method on geometrically conforming non-matching triangulations across the interface using the mortar technique. The resulting discrete problem is solved by a FETI-DP method. We prove that the method is convergent and its rate of convergence is almost optimal and independent of the jumps of coefficients. Numerical experiments for the case of four subregions are reported. They confirm the theoretical results.

Quadarant diagonal partitioning (QDP) method is an iterative method for solving linear system proposed by D. J. Evans et al. (see [1],[2],[3]) appropriate for paralell implementation. Here we show that in case of second order elliptic problem, the classical, color and multicolor ordering technique [4] are applicable for further paralellization of QDP iterative methods. Moreover we give a direct algorithm with arithmetical complexity O(n) for solving a special quadrant tridiagonal matrix and using these results we propose a quadrant tridiagonal preconditioned (QTP) conjugate gradient method for solving second order elliptic problems. The numerical experiments presented here shows that this method in several cases are more effective than other PCG algorithm.

BUILDING ELLIPTIC PROBLEM SOLVERS WITH ELLPACK

(1981). Second order elliptic boundary value problems in a domain with edges. Communications in Partial Differential Equations: Vol. 6, No. 9, pp. 1023-1042.

An economical cascadic multigrid method for the weak Galerkin finite element approximation of second order elliptic problems

A multi-domain method for solving numerically multi-scale elliptic problems

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well.

We develop and analyze a new discontinuous finite volume method for second order elliptic problems with tensor coefficients. An optimal order error estimate is obtained in a mesh dependent norm. An L2 -error estimate is also obtained.

In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second order self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the Raviart--Thomas and the Brezzi--Douglas--Marini methods of similar order are identical.