William F. Mitchell

A comparison of adaptive refinement techniques for elliptic problems

Adaptive refinement has proved to be a useful tool for reducing the size of the linear system of equations obtained by discretizing partial differential equations. We consider techniques for the adaptive refinement of triangulations used with the finite element method with piecewise linear functions. Several such techniques that differ mainly in the method for dividing triangles and the method for indicating which triangles have the largest error have been developed. We describe four methods for dividing triangles and eight methods for indicating errors. Angle bounds for the triangle division methods are compared. All combinations of triangle divisions and error indicators are compared in a numerical experiment using a population of eight test problems with a variety of difficulties (peaks, boundary layers, singularities, etc.). The comparison is based on the L-infinity norm of the error versus the number of vertices. It is found that all of the methods produce asymptotically optimal grids and that the number of vertices needed for a given error rarely differs by more than a factor of two.

A refinement-tree based partitioning method for dynamic load balancing with adaptively refined grids

The partitioning of an adaptive grid for distribution over parallel processors is considered in the context of adaptive multilevel methods for solving partial differential equations. A partitioning method based on the refinement-tree is presented. This method applies to most types of grids in two and three dimensions. For triangular and tetrahedral grids, it is guaranteed to produce connected partitions; no other partitioning method makes this guarantee. The method is related to the OCTREE method and space filling curves. Numerical results comparing it with several popular partitioning methods show that it computes partitions in an amount of time similar to fast load balancing methods like recursive coordinate bisection, and with mesh quality similar to slower, more optimal methods like the multilevel diffusive method in ParMETIS.

A Survey of hp-Adaptive Strategies for Elliptic Partial Differential Equations

The hp version of the finite element method (hp-FEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations because it can achieve a convergence rate that is exponential in the number of degrees of freedom. hp-FEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate can not simultaneously determine whether it is better to do the refinement by h or by p. Several strategies for making this determination have been proposed over the years. In this paper we summarize these strategies and demonstrate the exponential convergence rates with two classic test problems.

The full domain partition approach to distributing adaptive grids

Abstract Adaptive refinement has been shown to be an effective means of speeding up the solution of partial differential equations. Adaptive multilevel methods combine multigrid iteration with adaptive refinement to produce fast O(N) solutions on sequential computers, where N is the number of grid elements. However, many problems of interest require parallel computers. While adaptive refinement, multigrid and parallel computers are individually achieving widespread use in applications, the combination of all three is still a research topic. In this paper we describe the full domain partition (FuDoP) distribution of adaptively refined grids over a distributed memory parallel computer. In the FuDoP approach, each processor has a compatible grid that covers the full domain, but with refinement confined to the processors partition of the total grid. Outside the partition area, refinement is limited to that required for compatibility and leads to a small number of extra grid elements. With the FuDoP distribution, a parallel multigrid method can maintain a multigrid rate of convergence with only two communication steps per V-cycle. FuDoP also accommodates parallel implementation of adaptive refinement and partitioning.

A Comparison of hp -Adaptive Strategies for Elliptic Partial Differential Equations

The hp version of the finite element method (hp-FEM) combined with adaptive mesh refinement is a particularly efficient method for solving PDEs because it can achieve an exponential convergence rate in the number of degrees of freedom. hp-FEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate cannot simultaneously determine whether it is better to do the refinement by h or p. Several strategies for making this determination have been proposed over the years. These strategies are summarized, and the results of a numerical experiment to study the performance of these strategies is presented. It was found that the reference-solution-based methods are very effective, but also considerably more expensive, in terms of computation time, than other approaches. The method based on a priori knowledge is very effective when there are known point singularities. The method based on the decay rate of the expansion coefficients appears to be the best choice as a general strategy across all categories of problems, whereas many of the other strategies perform well in particular situations and are reasonable in general.

Analysis of monotectic growth: infinite diffusion in the L2 phase

The Jackson-Hunt model of eutectic solidification is applied to monotectic solidification in which a liquid (L1) transforms into rods of a different liquid (L2) in a solid matrix. Limiting cases of no diffusion and infinite diffusion (complete mixing) in the L2 phase are considered. An adaptive refinement and multigrid algorithm (MGGHAT) is used to obtain numerical solutions for the concentration field in the L1 phase; this allows consideration of a general phase diagram. Density differences between the three phases, which cause fluid flow, are treated approximately. Specific calculations are carried out for aluminum-indium alloys. Infinite diffusion in the L2 phase has only a small effect on the relationship between interface undercooling and rod spacing.

Effect of flow due to density change on eutectic growth

Abstract The Jackson–Hunt model of eutectic growth is extended to allow for different densities of the phases. The density differences give rise to fluid flow which is calculated from a series solution of the fluid flow equations in the Stokes flow approximation. The solute diffusion equation with flow terms is then solved numerically using an adaptive refinement and multigrid algorithm. The interface undercoolings and volume fractions are calculated as a function of spacing for tin–lead and iron–carbon eutectic alloys and for an aluminum–indium monotectic alloy. The numerical results are compared with various approximations based on the Jackson–Hunt analysis.

A collection of 2D elliptic problems for testing adaptive grid refinement algorithms

Adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations (PDEs). The development of new algorithms and computer codes for the solution of PDEs usually involves the use of proof-of-concept test problems. 2D elliptic problems are often used as the first test bed for new algorithms and codes. This paper contains a set of twelve parametrized 2D elliptic test problems for adaptive grid refinement algorithms and codes. The problems exhibit a variety of types of singularities, near singularities, and other difficulties.

The Full Domain Partition Approach to Parallel Adaptive Refinement

The combination of adaptive refinement, multigrid and parallel computing for solving partial differential equations is considered. In the full domain partition approach, each processor contains a partition of the grid plus the minimum number of additional coarse elements required to cover the whole domain. A parallel adaptive refinement algorithm using the full domain partition is presented. The method is a small modification of a sequential adaptive refinement algorithm, and uses no interprocessor communication during the refinement process. The only communication is one global reduction before refinement and three all-to-all communication steps for synchronization after the refinement is completed. Numerical computations on a network of up to 4 workstations show that parallel efficiency rates of 85% to near 100% can be obtained.

Effective-range description of a Bose gas under strong one- or two-dimensional confinement

We point out that theories describing s-wave collisions of bosonic atoms confined in one-dimensional (1D) or two-dimensional (2D) geometries can be extended to much tighter confinements than previously thought. This is achieved by replacing the scattering length by an energy-dependent scattering length which was already introduced for the calculation of energy levels under 3D confinement. This replacement accurately predicts the position of confinement-induced resonances in strongly confined geometries.