One-level Methods Research Articles

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

In this work we study the convergence properties of the one-level parallel Schwarz method with Robin transmission conditions applied to the one-dimensional and two-dimensional Helmholtz and Maxwell's equations. One-level methods are not scalable in general. However, it has recently been proven that when impedance transmission conditions are used in the case of the algorithm being applied to the equations with absorption, then, under certain assumptions, scalability can be achieved and no coarse space is required. We show here that this result is also true for the iterative version of the method at the continuous level for strip-wise decompositions into subdomains that are typically encountered when solving wave-guide problems. The convergence proof relies on the particular block Toeplitz structure of the global iteration matrix. Although non-Hermitian, we prove that its limiting spectrum has a near identical form to that of a Hermitian matrix of the same structure. We illustrate our results with numerical experiments.

A priori analysis of multilevel finite volume approximation of 1D convective Cahn–Hilliard equation

In this work, we analyze four finite volume methods for the nonlinear convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The methods used are: implicit one-level, explicit one-level, implicit multilevel and explicit multilevel finite volume methods. The existence and uniqueness of solution, convergence and stability of the finite volume solutions are proved. We compute $$L_2$$ - error and rate of convergence for all methods. We then compare the multilevel methods with the one-level methods by means of stability and CPU time. It is shown that the multilevel finite volume method is faster than the one-level method.

Two-level methods for the Cahn–Hilliard equation

We propose the fully discrete traditional finite element and mixed finite element two-level schemes for solving the Cahn–Hilliard equation in the paper. We give the stability and convergence of the traditional finite element and mixed finite element two-level methods. The analysis shows that the two-level methods can get the same convergence as the one-level methods provided that we choose proper coarse and fine mesh sizes. However, the two-level methods can save much computational time compared with the one-level methods. Finally, some numerical experiments are provided to confirm the theoretical analysis.

An efficient parallel algebraic multigrid method for 3D injection moulding simulation based on finite volume method

Elapsed time is always one of the most important performance measures for polymer injection moulding simulation. Solving pressure correction equations is the most time-consuming part in the mould filling simulation using finite volume method with SIMPLE-like algorithms. Algebraic multigrid (AMG) is one of the most promising methods for this type of elliptic equations. It, thus, has better performance by contrast with some common one-level iterative methods, especially for large problems. And it is also suitable for parallel computing. However, AMG is not easy to be applied due to its complex theory and poor generality for the large range of computational fluid dynamics applications. This paper gives a robust and efficient parallel AMG solver, A1-pAMG, for 3D mould filling simulation of injection moulding. Numerical experiments demonstrate that, A1-pAMG has better parallel performance than the classical AMG, and also has algorithmic scalability in the context of 3D unstructured problems.

Performance Analysis of Two-Level Asynchronous Optical CDMA Systems Utilizing Wrapped OPPM

In this paper the performance and the capacity of two-level asynchronous incoherent optical CDMA (OCDMA) systems employing wrapped overlapping pulse position modulation (WOPPM) and optical AND logic gate receiver are analyzed. These analyses are performed by considering the multiple access interference as the main limiting factor. Because of the power limitation in some laser sources, the same-symbol-power scheme for all users is assumed. Optical orthogonal codes with the same length are assigned to the users of each class. Considering the throughput as constraint, the effects of the number of users in each class and the weight of low-weight users on the performance of each class are examined and the results are compared with the same-mark-power case. Finally, the effect of WOPPM signaling parameters such as overlapping index and total number of symbols on the error probability of each class is investigated in the presence and in the absence of throughput constraint. Our results show that when the multi-level technique is employed in WOPPM OCDMA systems, one can find solutions for some design problems that cannot be handled by conventional one-level methods. In addition to this benefit, employing this technique leads to a less complex transceiver structure.

Local Projection Stabilization Method for Incompressible Flow Problems

We analyze Local Projection Stabilization (LPS) methods for the solution of Stokes problem using equal order finite elements. We investigate their convergence, stability and accuracy properties. The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations. We distinguish two classes of LPS methods: one-level and two-level methods. Numerical examples using bilinear interpolations are presented to validate the analysis and assess the accuracy of both approaches.

A two-grid finite difference method for the primitive equations of the ocean

A two-level method is proposed for the finite difference approximation of a class of dissipative systems including the primitive equations of the ocean. The two-level (or multilevel) method amounts to solving a nonlinear problem on the single coarsest grid; subsequent approximations are generated on a succession of refined meshes by solving a linearized problem about the solution on the previous level. The stability analysis for an Euler-type scheme shows that the multilevel method can allow larger time steps than the traditional one-level methods. Some numerical simulations of the primitive equations of the ocean using this new multi-scale algorithm are presented to illustrate the method.

Multidisciplinary design optimisation techniques

Design optimisation of a multidisciplinary project in engineering involves the decomposition of a system into disciplines and the subsequent association of their contributions. This work was aimed at presenting the most common decomposition and association techniques currently used in multidisciplinary design optimisation (MDO). Amongst the decomposition techniques this work includes hierarchical and non-hierarchical approaches as well as the most popular numerical procedures. The association techniques include: one-level methods (e.g. all-at-once optimisation and simultaneous analysis and design), multilevel methods (e.g. concurrent subspace optimisation and collaborative optimisation) and robust design. This work also incorporates an illustrative numerical example.

A review of algebraic multigrid

Since the early 1990s, there has been a strongly increasing demand for more efficient methods to solve large sparse, unstructured linear systems of equations. For practically relevant problem sizes, classical one-level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems. This paper gives a review of the first hierarchical and purely matrix-based approach, algebraic multigrid (AMG). AMG can directly be applied, for instance, to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes, both in 2D and 3D. Since AMG does not make use of any geometric information, it is a “plug-in” solver which can even be applied to problems without any geometric background, provided that the underlying matrix has certain properties.

Hierarchical control of biotechnological complex systems

Application of hierarchical control for solution of some nonlinear optimization problems in fermentation systems, with the use of microprocessors, is described. Results presented show some advantages of the method of hierarchical control in comparison to some traditional one-level optimization methods. The advantage of the hierarchical control method is in less demand on memory and computing power of the control computer as the standard methods. In solving this problem two control schemes are used. In the first control scheme the method of objective coordination is used. In the second one the prediction method of coordination is used, with the aim to minimize selected disturbances in state and control variables.