Yuri A. Kuznetsov

Convergence analysis and error estimates for mixed finite element method on distorted meshes

In [2] we introduced a new type of mixed finite element approximations for two- and three-dimensional problems on distorted polygonal and polyhedral meshes that consist of cells having different forms. Additional degrees of freedom that arise in the process are excluded by a special condition that is natural for the mixed finite element approximations considered. This paper is devoted to the error analysis of the respective finite element solutions. We show that under certain assumptions on the regularity of the exact solution the convergence rate for the new approximations is the same as for the Raviart–Thomas finite element approximations of the lowest order.

FICTITIOUS DOMAIN METHODS FOR THE NUMERICAL SOLUTION OF THREE-DIMENSIONAL ACOUSTIC SCATTERING PROBLEMS

Efficient iterative methods for the numerical solution of three-dimensional acoustic scattering problems are considered. The underlying exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a non-reflecting boundary condition on the artificial boundary. The finite element discretization of the approximate boundary value problem is performed using locally fitted meshes, and algebraic fictitious domain methods with separable preconditioners are applied to the solution of the resultant mesh equations. These methods are based on imbedding the original domain into a larger one with a simple geometry (for example, a sphere or a parallelepiped). The iterative solution method is realized in a low-dimensional subspace, and partial solution methods are applied to the linear systems with the preconditioner. The results of numerical experiments demonstrate the efficiency and accuracy of the approach.

Mixed finite element method for diffusion equations on polygonal meshes with mixed cells

In this paper, a new mixed finite element method for the diffusion equation on polygonal meshes is proposed. The method is applied to the diffusion equation on meshes with mixed cells when all the coefficients and the source function may have discontinuities inside polygonal mesh cells. The resulting discrete equations operate only with the degrees of freedom for normal fluxes on the boundaries of cells and one degree of freedom per cell for the solution function.

Guaranteed lower bounds of the smallest eigenvalues of elliptic differential operators

Abstract This paper presents a rigorous analysis of the method of computing guaranteed lower bounds of the smallest eigenvalue of an elliptic operator in the case of mixed or purely Neumann boundary conditions. The method was originally invented in [8]. It is based on a decomposition of a domain into a set of overlapping subdomains, for which the corresponding estimates of minimal positive eigenvalues are known or easily computable.We prove that finding a guaranteed lower bound can be reduced to a finite dimensional variational problem. The dimension of the problem depends on the amount of subdomains, and the structure of the corresponding functional depends on topological properties of the set of overlapping subdomains. Several examples show the performance of the estimates.

Approximations with piece-wise constant fluxes for diffusion equations

Abstract — In this paper, a new discretization method for the diffusion equation in the m ix d formulation is proposed. This method is applied to the discretization of diffus ion on polygonal (2D) and polyhedral (3D) meshes. The method is based on the approx imation of the flux vector function in mesh cells by piece-wise constant (PWC) vector functions. In a particular case the error estimate for the flux vector function is derived.

Preconditioned iterative methods for algebraic saddle-point problems

Abstract In this paper, we consider the preconditioned Lanczos method for the numerical solution of algebraic systems with singular saddle point matrices. These systems arise from algebraic systems with singularly perturbed symmetric positive definite matrices. The original systems are replaced by equivalent systems with saddle point matrices. Two approaches are proposed to design preconditioners for singular saddle point matrices. The algorithms are applied to the diffusion equation with strongly heterogeneous and anisotropic diffusion tensors.

Fast separable solver for mixed finite element methods and applications

Abstract A separable preconditioner for the solution of elliptic equations in the mixed-hybrid form is proposed and studied. The problem is discretized on logically rectangular (d = 2 or 3) meshes with the lowest order Raviart–Thomas finite elements. The arithmetical complexity of the preconditioner is O(nln d–1 n). Results of numerical experiments are presented.