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Open Access
January 1, 2003
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Open Access
January 1, 2003
### Abstract

Preconditioners based on various multilevel extensions of two-level piecewise linear finite element methods lead to iterative methods which have an optimal order computational complexity with respect to the size (or discretization parameter) of the system. The methods can be in block matrix factorized form, recursively extended via certain matrix polynomial approximations of the arising Schur complement matrices or on additive, i.e., block diagonal form using stabilizations of the condition number at certain levels. The resulting spectral equivalence holds uniformly with respect to jumps in the coefficients of the differential operator and for arbitrary triangulations. Such methods were first presented by Axelsson and Vassilevski in the late 1980s. An important part of the algorithm is the treatment of systems with a diagonal block matrix, which arises on each finer level in a recursive refinement method and corresponds to the added degrees of freedom on that level. This block is well-conditioned for model type problems but becomes increasingly ill-conditioned when the coefficient matrix becomes more anisotropic or, equivalently, when the mesh aspect ratio increases. This paper presents some methods for approximating this matrix also leading to a preconditioner with spectral equivalence bounds which hold uniformly with respect to both the problem and the discretization parameters. Therefore, the same holds also for the preconditioner to the global matrix.

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Open Access
January 1, 2003
### Abstract

In this paper, we present a non-conforming hp computational modeling methodology for solving elasticity problems. We consider the incompressible elasticity model formulated as a mixed displacement-pressure problem on a global domain which is partitioned into several local subdomains. The approximation within each local subdomain is designed using div-stable hp-mixed finite elements. The displacement is computed in a mortared space while the pressure is not subjected to any constraints across the interfaces. Our computational results demonstrate that the non-conforming finite element method presented for the elasticity problem satisfies similar rates of convergence as the conforming finite element method, in the presence of various h-version and p-version discretizations.

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Open Access
January 1, 2003
### Abstract

In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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Open Access
January 1, 2003
### Abstract

In this paper we derive error estimates for finite element approximations for partial differential systems which describe two-phase immiscible flows in porous media. These approximations are based on mixed finite element methods for pressure and velocity and characteristic finite element methods for saturation. Both incompressible and compressible flows are considered. Error estimates of optimal order are obtained.

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Open Access
January 1, 2001
### Abstract

A boundary-value problem with a nonlocal integral condition is considered for a two-dimensional elliptic equation with constant coefficients and a mixed derivative. The existence and uniqueness of a weak solution of this problem are proved in a weighted Sobolev space. A difference scheme is constructed using the Steklov averaging operators.

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Open Access
January 1, 2003
### Abstract

Discontinuous Galerkin methods for elliptic problems with discontinuous coefficients are discussed. First the error bound of the methods is analyzed. Then a multilevel additive Schwarz preconditioner for one of the discrete problems is designed and analyzed. Although the preconditioner is not optimal it is very well suited for parallel computations and its rate of convergence is independent of the jumps of coefficients.

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Open Access
January 1, 2003
### Abstract

We have developed a discretization method of an arbitrarily given order of accuracy with respect to the time discretization parameter for the first-order evolution differential equation in Banach space. The method includes two levels of parallelism: the operator exponential needed for the calculation of the evolution operator can be computed in parallel (inner parallelism) and then we can compute in parallel the evolution operator at various points of the time mesh (outer parallelism).

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Open Access
January 1, 2003
### Abstract

In this paper, we provide a Schwarz preconditioner for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods. The preconditioner is for the linear equation for Lagrange multipliers arrived at by eliminating the flux as well as the primal variable. We also prove a condition number estimate for this equation when no preconditioner is used. Although preconditioners for the lowest-order case of the Raviart-Thomas method have been constructed previously by exploiting its connection with a nonconforming method, our approach is different in that we use a new variational characterization of the Lagrange multiplier equation. This allows us to precondition even the higher-order cases of these methods.

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Open Access
January 1, 2003
### Abstract

The purpose of this paper is to introduce discretization methods of discontinuous Galerkin type for solving second-order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary. Starting with the Lagrange multiplier formulation for the boundary conditions, we derive variational forms for the discretization of 2-D elliptic problems with embedded Dirichlet boundary conditions. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DGdiscretization is adapted in cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection-dominated boundary-value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of cubic polynomials, the boundary condition treatment appears quite effective in handling such complex situations.

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Open Access
January 1, 2003
### Abstract

We are concerned with structural optimization problems for technological processes in material science that are described by partial differential equations. In particular, we consider the topology optimization of conductive media in high-power electronic devices described by Maxwell equations and the optimal design of composite ceramic materials by homogenization modeling. All these tasks lead to constrained nonconvex minimization problems with both equality and inequality constraints on the state variables and design parameters. After discretization by finite elements, we solve the discretized optimization problems by a primal-dual Newton interior-point method. Within a line-search approach, transforming iterations are applied with respect to the null space decomposition of the condensed primal-dual system to find the search direction. Some numerical experiments for the two applications are presented.

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Open Access
January 1, 2003
### Abstract

The convergence of difference schemes for the two–dimensional weakly parabolic equation (elliptic equation with a dynamic interface condition) is studied. Estimates for the rate of convergence “almost” (except for the logarithmic factor) compatible with the smoothness of the differential problem solution in special discrete Sobolev norms are obtained.

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Open Access
January 1, 2003
### Abstract

A new mixed finite volume method for elliptic equations with tensor coefficients on rectangular meshes (2 and 3-D) is presented. The implementation of the discretization as a finite volume method for the scalar variable (“pressure”) is derived. The scheme is well suited for heterogeneous and anisotropic media because of the generalized harmonic averaging. It is shown that the method is stable and well posed. First-order error estimates are derived. The theoretical results are confirmed by the presented numerical experiments.

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Open Access
January 1, 2003
### Abstract

We consider the Poisson equation with Dirichlet boundary conditions on a polygonal domain with one reentrant corner. We introduce new nonconforming finite element discretizations based on mortar techniques and singular functions. The main idea introduced in this paper is the replacement of cut-off functions by mortar element techniques on the boundary of the domain. As advantages, the new discretizations do not require costly numerical integrations and have smaller a priori error estimates and condition numbers. Based on such an approach, we prove optimal accuracy error bounds for the discrete solution. Based on such techniques, we also derive new extraction formulas for the stress intensive factor. We establish optimal accuracy for the computed stress intensive factor. Numerical examples are presented to support our theory.