Nodal methods : analysis, performance and fast iterative solvers

Nodal Methods have long been one of the most popular discretization techniques employed within the reactor physics community, while remaining conspicuously absent from the mainstream numerical analysis literature. A fundamental reason for this anomaly is that the physical arguments which were used to develop and enhance these methods seemed at odds with more rigorous discretization techniques. To facilitate communication between these distinct communities, a detailed chronological study of the lowest-order nodal methods for linear second order elliptic problems is presented. The presentation highlights the underlying motivation of these methods and formalizes many of their renowned properties. In addition, various equivalence relations within this family of discretizations are demonstrated, and equivalences with specific non-conforming and mixed-hybrid finite element methods (FEMs) are established. Rigorous error bounds and stability properties follow immediately from these latter equivalence relations, corroborating the results of a more rudimentary truncation error analysis. An inherent difficulty of reactor simulation is that the coefficients in the mathematical model exhibit severe variations on two significantly different length scales. As in many other applications this is treated by defining an appropriate homogenization procedure which yields a simplified model with piecewise constant coefficients on a coarse scale suitable for efficient computation. Significant enhancements in accuracy are possible if the processes of homogenization and discretization are unified. We review the popular techniques that are based on this premise and rely on certain properties of the nodal discretization. In addition, we address the factors that contribute to their success in reactor modelling and deter their generalization outside of the reactor physics community. As an alternative to these highly specialized methods, we introduce a new multi-level homogenization technique which is readily applicable in a general setting, and is shown to have many important attributes. Widespread acceptance of nodal methods has also been hindered by their use of nonstandard unknowns, as this results in stencils that appear awkward and incompatible with sophisticated iterative solution techniques. Specifically, equivalence with certain mixed-hybrid FEMs reveals that the nodal discretizations result in an indefinite system, which in two dimensions contains both cell-based and edge-based unknowns. Yet, inherent in this structure is a natural partitioning of the system which may be exploited to define a hierarchy of reduced systems (i.e. Schur complements) that are symmetric positive definite. Unfortunately, the reduced systems that involve unknowns of only one type, and hence seem most compatible with sophisticated iterative methods, suffer a loss of sparsity. However, the structure inherent in this hierarchy may be utilized in the construction of sparse approximate Schur complements for these systems. It is demonstrated that any one of these approximate operators, which are of either the standard 5 or 9-point family, may be utilized as an excellent preconditioner for conjugate gradient iterations. The efficiency of this approach is fully realized when the preconditioner is approximately inverted using only a single V - or W-cycle of a robust Black Box multigrid solver.

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