In this paper, we consider the numerical solution of the optimal control problems of the elliptic partial differential equation. Numerically tackling these problems using the finite element method produces a large block coupled algebraic system of equations of saddle point form. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The solution of such systems is a major computational task and poses a greater challenge for iterative techniques. Thus they require specialised methods which involve some preconditioning strategies. The preconditioned solvers must have nice convergence properties independent of the changes in discretisation and problem parameters. Most well known preconditioned solvers converge independently of mesh size but not for the decreasing regularisation parameter. This work proposes and extends the work for the formulation of preconditioners which results in the optimal performances of the iterative solvers independent of both the decreasing mesh size and the regulation parameter. In this paper we solve the indefinite system using the preconditioned minimum residual method. The main task in this work was to analyse the 3 × 3 block diagonal preconditioner that is based on the approximation of the Schur complement form obtained from the matrix system. The eigenvalue distribution of both the proposed Schur complement approximate and the preconditioned system will be investigated since the clustering of eigenvalues points to the effectiveness of the preconditioner in accelerating an iterative solver. This is done in order to create fast, efficient solvers for such problems. Numerical experiments demonstrate the effectiveness and performance of the proposed approximation compared to the other approximations and demonstrate that it can be used in practice. The numerical experiments confirm the effectiveness of the proposed preconditioner. The solver used is robust and optimal with respect to the changes in both mesh size and the regularisation parameter.

The study considers the saddle point problem arising from the mixed finite element discretization of the steady state Stokes equations. The saddle point problem is an indefinite system of linear equations, a feature that degrades the performance of any iterative solver. The heart of the study is the construction of fast, robust and effective iterative solution methods for such systems. Specific attention is given to the preconditioned MINRES solver PMINRES which is carefully treated for the solution of the Stokes equations. The study concentrates on the block preconditioner applied to the MINRES to effectively solve the whole coupled system. We combine iterative techniques with the MINRES as preconditioner approximations to produce an efficient solver for indefinite system of equations. We consider different preconditioner approximations of the building blocks of the preconditioner and compare their effects in accelerating the MINRES iterative scheme. We give a detailed overview of the algorithmic aspects and the theoretical convergence analysis of our solver. We study the MINRES method with the following preconditioner approximations: diagonal, multigrid v-cycle, preconditioned conjugate gradient and Chebyshev semi iteration methods. A comparative analysis of the preconditioner approximations show that the multigrid method is a suitable accelerator for the MINRES method. The application of the preconditioner becomes mandatory as evidenced by poor performance of the MINRES as compared to PMINRES. We study the problem in a two dimensional setting using the Hood-Taylor Q 2 − Q 1 stable pair of finite elements. The incompressible flow iterative solution software(IFISS) matlab toolbox is used to assemble the matrices. We present the numerical results to illustrate the efficiency and robustness of the MINRES scheme with the multigrid preconditioner.

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q 2-Q 1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results.

In this study, the optimal control problem is considered. This is an important class of partial differential equations constrained optimization problems. The constraint here is an elliptic partial differential equation with Neumann boundary conditions. The discretization of the optimality system produces a block coupled algebraic system of equations of saddle point form. The solution of such systems is a major computational task since they require specialized methods. Constructing robust, fast and efficient solvers for their numerical solution has preoccupied the computational science community for decades and various approaches have been developed. The approaches involve solving simultaneously for all the unknowns using a coupled block system, the segregated approach where a reduced system is solved and the approach of reducing to a fixed point form. Here the minimum residual solver with ideal preconditioning is applied to the unreduced 3 by 3 and reduced 2 by 2 coupled systems and compared to the multigrid method applied to the compact fixed-point form. The two methods are compared numerically in terms of iterative counts and computational times. The numerical results indicate that the two methods produce similar outcomes and the multigrid solver becoming very competitive in terms of the iterative counts though slower than preconditioned minimum residual solver in terms of computational times. For all the approaches, the two methods exhibited mesh and parameter independent convergence. The optimal performance of the two methods is verified computationally and theoretically.

This paper investigates the optimal cash crop combination at a small-scale cash crop farm in Vhembe District, Limpopo Province, South Africa in which green maize, cabbages, tomatoes, spinach, mustard, butternut and sweet potatoes are grown. To get optimum farm outputs, decisions on crop combination and operational activities in crop production are crucial. Proper farm planning and resource allocation play a significant role in optimising farm revenues. It was observed that the farmer used traditional methods of allocating resources, which lead to a less profitable crop mix. In view of this, in this study, linear programming model was formulated using data collected from a farm concerning the past crop combinations and allocation of resources in crop planning and production to determine the best crop combination that maximizes net income given limited resources such as land, labour, capital, and others. The simplex method of linear programming works by first locating a feasible solution and then relocating to any vertex of the feasible set that improves the cost function. Eventually, a point is reached beyond which no further movement improves the cost function. The results of the developed linear programming model were compared to past farming practices based on experience, leading to the conclusion that crops and limited resources were not optimally allocated. The results clearly demonstrate the optimal crop combination and allocation of scarce resources that the farm could have considered to yield maximum returns. It is observed that the proposed linear programming model is appropriate for finding the optimal land allocation criteria for the cash crops in the study area. The optimal crop mix from the linear programming two phase simplex method show that the farmer should grow the following crop mix, 1.16 ha of green mealies, 2.64 ha of cabbages, 0.8 ha of tomatoes, 1.2 ha of mustard, 0.4 ha spinach, 0.4 ha of butternut and 0.4 ha of sweet potatoes with the gross income of R 740 800. The linear programming model resulted in a 37.8 percent increase in profit margin. Based on the results obtained from this study, it is recommended that the small-scale cash crop farmer should invest more in producing crops that give maximum profit. MATLAB software was used to determine the optimal values of the decision variables. Key words: Cash crops allocation, crop combination, net income, linear programming (LP), resource allocation.