An inexact Newton method for systems arising from the finite element method

In this paper, we introduce an efficient and robust technique for approximating the Jacobian matrix for a nonlinear system of algebraic equations which arises from the finite element discretization of a system of nonlinear partial differential equations. It is demonstrated that when an iterative solver, such as preconditioned GMRES, is used to solve the linear systems of equations that result from the application of Newton's method, this approach is generally more efficient than using matrix-free techniques: the price paid being the extra memory requirement for storing the sparse Jacobian. The advantages of this approach over attempting to calculate the Jacobian exactly or of using other approximations are also discussed. A numerical example is included which is based upon the solution of a 2-d compressible viscous flow problem

Parallel performance prediction for multigrid codes on distributed memory architectures

We propose a model for describing the parallel performance of multigrid software on distributed memory architectures. The goal of the model is to allow reliable predictions to be made as to the execution time of a given code on a large number of processors, of a given parallel system, by only benchmarking the code on small numbers of processors. This has potential applications for the scheduling of jobs in a Grid computing environment where reliable predictions as to execution times on different systems will be valuable. The model is tested for two different multigrid codes running on two different parallel architectures and the results obtained are discussed

A weakly overlapping parallel domain decomposition preconditioner for the finite element solution of convection-dominated problems in three dimensions

In this paper we describe the parallel application of a novel two level additive Schwarz preconditioner to the stable finite element solution of convection-dominated problems in three dimensions. This is a generalization of earlier work, [2,6], in 2-d and 3-d respectively. An algebraic formulation of the preconditioner is presented and the key issues associated with its parallel implementation are discussed. Some computational results are also included which demonstrate empirically the optimality of the preconditioner and its potential for parallel implementation

A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification

A fully-implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2D is presented. In addition we combine a second-order fully-implicit time discretisation scheme with variable steps size control to obtain an adaptive time and space discretisation method. The superiority of this method, compared to widely used fully-explicit methods, with respect to CPU time and accuracy, is shown. Due to the high non-linearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step. We use a nonlinear multigrid solver which shows almost h-independent convergence behaviour

An adaptive, fully implicit multigrid phase-field model for the quantitative simulation of non-isothermal binary alloy solidification

Using state-of-the-art numerical techniques, such as mesh adaptivity, implicit time-stepping and a non-linear multi-grid solver, the phase-field equations for the non-isothermal solidification of a dilute binary alloy have been solved. Using the quantitative, thin-interface formulation of the problem we have found that at high Lewis number a minimum in the dendrite tip radius is predicted with increasing undercooling, as predicted by marginal stability theory. Over the dimensionless undercooling range 0.2–0.8 the radius selection parameter, σ*, was observed to vary by over a factor of 2 and in a non-monotonic fashion, despite the anisotropy strength being constant

Anisotropic adaptivity for the finite element solutions of three-dimensional convection-dominated problems

Convection-dominated problems are typified by the presence of strongly directional features such as shock waves or boundary layers. Resolution of numerical solutions using an isotropic mesh can lead to unnecessary refinement in directions parallel to such features. This is particularly important in three dimensions where the grid size increases rapidly during conventional isotropic refinement procedures. In this work, we investigate the use of adaptive finite element methods using anisotropic mesh refinement strategies for convection-dominated problems. The strategies considered here aim to resolve directional features without excessive resolution in other directions, and hence achieve accurate solutions more efficiently. Two such strategies are described here: the first based on minimization of the least-squares residual; the second based on minimizing a finite element error estimate. These are incorporated into an hr-adaptive finite element method and tested on a simple model problem

Advanced numerical methods for the simulation of alloy solidification with high Lewis number

A fully-implicit numerical method based upon adaptively refined meshes for the thermal-solutal simulation of alloy solidification in 2D is presented. In addition we combine an unconditional stable second-order fully-implicit time discretisation scheme with variable step size control to obtain an adaptive time and space discretisation method, where a robust and fast multigrid solver for systems of non-linear algebraic equations is used to solve the intermediate approximations per time step. For the isothermal case, the superiority of this method, compared to widely used fully-explicit methods, with respect to CPU time and accuracy, has been demonstrated and published previously. Here, the new proposed method has been applied to the thermalsolutal case with high Lewis number, where stability issues and time step restrictions have been major constraints in previous research

Consistent dirichlet boundary conditions for numerical solution of moving boundary problems

We consider the imposition of Dirichlet boundary conditions in the finite element mod-elling of moving boundary problems in one and two dimensions for which the total mass is prescribed. A modification of the standard linear finite element test space allows the boundary conditions to be imposed strongly whilst simultaneously conserving a discrete mass. The validity of the technique is assessed for a specific moving mesh finite element method, although the approach is more general. Numerical comparisons are carried out for mass-conserving solutions of the porous medium equation with Dirichlet boundary conditions and for a moving boundary problem with a source term and time-varying mass

Improved parallel mesh generation through dynamic load-balancing

Parallel mesh generation is an important feature of any large distributed memory parallel computational mechanics code due to the need to ensure that (i) there are no sequential bottlenecks in the code, (ii) there is no parallel overhead incurred in partitioning an existing mesh, and (iii) that no single processor is required to have enough local memory to be able to store the entire mesh

A multilevel approach for obtaining locally optimal finite element meshes

In this paper we consider the adaptive finite element solution of a general class of variational problems using a combination of node insertion, node movement and edge swapping. The adaptive strategy that is proposed is based upon the construction of a hierarchy of locally optimal meshes starting with a coarse grid for which the location and connectivity of the nodes is optimized. This grid is then locally refined and the new mesh is optimized in the same manner. Results presented indicate that this approach is able to produce better meshes than those possible by more conventional adaptive strategies and in a relatively efficient manner