Mixed Finite Element Discretization Research Articles

Balancing domain decomposition for mixed finite elements

The rate of convergence of the Balancing Domain Decomposition method applied to the mixed finite element discretization of second-order elliptic equations is analyzed. The Balancing Domain Decomposition method, introduced recently by Mandel, is a substructuring method that involves at each iteration the solution of a local problem with Dirichlet data, a local problem with Neumann data, and a "coarse grid" problem to propagate information globally and to insure the consistency of the Neumann problems. It is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size, in both two and three dimensions and for elements of arbitrary order. The bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains. The key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the H 1 / 2 {H^{1/2}} -norm of an interpolant of the boundary data. Computational results from a message-passing parallel implementation on an INTEL-Delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed-up for up to 64 processors.

Inexact and Preconditioned Uzawa Algorithms for Saddle Point Problems

Variants of the Uzawa algorithm for solving symmetric indefinite linear systems are developed and analyzed. Each step of this algorithm requires the solution of a symmetric positive-definite system of linear equations. It is shown that if this computation is replaced by an approximate solution produced by an arbitrary iterative method, then with relatively modest requirements on the accuracy of the approximate solution, the resulting inexact Uzawa algorithm is convergent, with a convergence rate close to that of the exact algorithm. In addition, it is shown that preconditioning can be used to improve performance. The analysis is illustrated and supplemented using several examples derived from mixed finite element discretization of the Stokes equations.

Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics

Block matrices with a special structure arise from mixed finite element discretizations of incompressible flow problems. This paper is concerned with an analysis of the eigenvalue problem for such matrices and the derivation of two shifted eigenvalue problems that are more suited to numerical solution by iterative algorithms like simultaneous iteration and Arnoldi's method. The application of the shifted eigenvalue problems to the determination of the eigenvalue of smallest real part is discussed and a numerical example arising from a stability analysis of double-diffusive convection is described.

Iterative solution techniques for the stokes and Navier‐Stokes equations

AbstractThe linear system arising from a Lagrange‐Galerkin mixed finite element approximation of the Navier–Stokes and continuity equations is symmetric indefinite and has the same block structure as a system arising from a mixed finite element discretization of a Stokes problem. This paper considers the iterative solution of such a system, comparing the performance of the one‐level preconditioned conjugate residual method for indefinite matrices with that of a more traditional two‐level pressure correction approach. Asymptotic estimates for the amount of work involved in each method are given together with the results of related numerical experiments.

Modeling and simulation of solid-state transducers: the thermal and electrical energy domain

Abstract This paper deals with a basic strategy towards the (efficient) numerical modeling of semiconductor micro-structures in which electric, thermal and thermoelectric effects are important. In particular, we discuss a very efficient method of O(n) for solving the nonlinear semiconductor thermoelectric model equations, where n is the total number of elements in the grid. The method is based on the multigrid method in combination with the mixed finite element discretization method and a nonlinear Vanka-type relaxation method. For a simple, but representative test problem we demonstrate the feasibility of this approach by showing that (nearly) mesh-size independent convergence rates can be obtained.

Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part I: Algorithms and numerical results

We describe sequential and parallel algorithms based on the Schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the Raviart-Thomas finite element spaces. These lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block Gauss-Seidel or block Jacobi methods with overlapping blocks. The indefiniteness requires special treatment. The sub-blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of Dryja and Widlund for standard elliptic problems. If there is sufficient overlap between the subdomains, the algorithm converges with a rate independent of the mesh size, the number of subdomains and discontinuities of the coefficients. Extensions of the above algorithms to the case of local grid refinement is also described. Convergence theory for these algorithms will be presented in a subsequent paper.

Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part II: Convergence theory

In this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric, indefinite linear systems arising from mixed finite element discretizations of elliptic problems. The algorithms include Schwarz methods and iterative refinement methods on locally refined grids. The implementation of Schwarz and iterative refinement algorithms have been discussed in part I. A discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived.

Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations

This paper is concerned with a numerical approach to the problem of finding the leftmost eigenvalues of large sparse nonsymmetric generalised eigenvalue problems which arise in stability studies of incompressible fluid flow problems. The matrices have a special block structure that is typical of mixed finite element discretizations for such problems. The numerical approach is an extension of the hybrid technique introduced by Saad [22] and utilizes the idea of preconditioning the eigenvalue problem before applying Arnoldi's method. Two preconditioners, one a modified Cayley transform, the other a Chebyshev polynomial transform, are compared in numerical experiments on a double diffusive convection problem and the Cayley transform proves superior. The Cayley transform is then used to provide numerical results for the finite Taylor problem.

A modified Cayley transform for the discretized Navier-Stokes equations

This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalized eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem. Numerical experiments are performed on large matrices arising from a discretization of the flow over a backward facing step.

Substructure preconditioners for elliptic saddle point problems

Domain decomposition preconditioners for the linear systems arising from mixed finite element discretizations of second-order elliptic boundary value problems are proposed. The preconditioners are based on subproblems with either Neumann or Dirichlet boundary conditions on the interior boundary. The preconditioned systems have the same structure as the nonpreconditioned systems. In particular, we shall derive a preconditioned system with conditioning independent of the mesh parameter h. The application of the minimum residual method to the preconditioned systems is also discussed.

Multilevel iterative methods for mixed finite element discretizations of elliptic problems

For solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable.

MIXED FINITE ELEMENT BASED NEURAL NETWORKS IN VISUAL RECONSTRUCTION

This paper shows how visual reconstruction problems can be solved using analog networks resulting from the reformulation of the reconstruction problem into a series of coupled sub-problems. A novel analog implementation based upon Platt's constraint networks is suggested for neural network implementation and the effectiveness is demonstrated with examples. The complete approach bears a philosophical similarity to the Harris Coupled Depth-Slope model of visual reconstruction. Significant differences appear, though, in the Lagrangian formulation, the mixed finite element discretization, and the analog implementation. It is also shown that the resulting networks can be structurally different from those suggested by Harris.

Vectorizable preconditioners for mixed finite element solution of second-order elliptic problems

We compare the performance of the CG (conjugate gradient) method, the point-wise incomplete factorization preconditioned CG method, and the block-incomplete factorization preconditioned CG method for solving problems arising in mixed finite element discretization of second order elliptic differential equations. The robustness and vectorizable properties of these methods are illustrated by a large set of numerical tests.

Computing material collapse displacement fields on a cray X-MP/48 by the LP primal affine scaling algorithm

A pair of dual, highly degenerate linear programs modeling the static and kinematic principles of limit analysis are studied. They arise from mixed finite element discretizations of the continuous saddle point problem and piecewise linear approximations of the convex unbounded Mises set of admissible stresses. We use piecewise constant stresses and piecewise bilinear displacements, and present computational results with the Affine Scaling Algorithm a la Dikin (1967) and Karmarker (1984). Graphical displays are presented for material collapse fields of a rectangular solid with thin cuts under tension, in problem sizes up to 9000 variables and 7700 equations. To our knowledge, these stress fields have not been computed before.

A discourse on the stability conditions for mixed finite element formulations

We discuss the general mathematical conditions for solvability, stability and optimal error bounds of mixed finite element discretizations. Our objective is to present these conditions with relatively simple arguments. We present the conditions for solvability and stability by considering the general coefficient matrix of mixed finite element discretizations, and then deduce the conditions for optimal error bounds for the distance between the finite element solutions and the exact solution of the mathematical problem. To exemplify our presentation we consider the solutions of various example problems. Finally, we also present a numerical test that is useful to identify numerically whether, for the solution of the general Stokes flow problem, a given finite element discretization satisfies the stability and optimal error bound conditions.

A modified method of characteristics technique and mixed finite elements method for simulation of groundwater solute transport

A comprehensive groundwater solute transport simulator is developed based on the modified method of characteristics (MMOC) combined with the Galerkin finite element method for the transport equation and the mixed finite element (MFE) method for the groundwater flow equation. The preconditioned conjugate gradient algorithm is used to solve the two large sparse algebraic system of equations arising from the MMOC and MFE discretizations. The MMOC takes time steps in the direction of flow, along the characteristics of the velocity field of the total fluid. The physical diffusion and dispersion terms are treated by a standard finite element scheme. The crucial aspect of the MMOC technique is that it looks backward in time, along an approximate flow path, instead of forward in time as in many method of characteristics or moving mesh techniques. The MFE procedure involves solving for both the hydraulic head and the specific discharge simultaneously. One order of convergence is gained by the MFE method, as compared with other standard finite element methods, and therefore more accurate velocity fields are simulated. The overall advantages of the MMOC‐MFE method include minimum numerical oscillation or grid orientation problems under steep concentration gradient simulations, and material balance errors are greatly reduced due to a very accurate velocity simulation by the MFE method. In addition, much larger time steps with Courant number well in excess of 1, as compared with the standard Galerkin finite element method, can be taken on a fixed spatial grid system without significant loss of accuracy.

Computations in limit analysis for plastic plates

AbstractA class of mixed finite element discretizations of the problem of limit analysis for plastic plates is analysed. For the discrete problem we discuss a linear programming approach based on linearization of the yield condition and a convex programming approach based on the exact Mises condition. Applying the method to rectangular plates with uniform load and concentrated loads we get improved and new results partly in disagreement with one of our references.

A mixed finite element solution of some biharmonic unilateral problem

We consider some plate obstacle problem and apply a mixed finite element discretization method. Finally we draw some algorithm for solving the finite-dimensional complementarity systems so obtained.

A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator

AbstractThe homogeneous Dirichlet problem for the biharmonic operator is solved as the variational formulation of two coupled second‐order equations. The discretization by a mixed finite element model results in a set of linear equations whose coefficient matrix is sparse, symmetric but indefinite. We describe a class of preconditioned conjugate gradient methods for the numerical solution of this linear system. The precondition matrices correspond to incomplete factorizations of the coefficient matrix. The numerical results show a low computational complexity in both number of computer operations and demand of storage.