Adaptive methods for finite volume approximations

In this dissertation we construct, theoretically justify, and test computational methods and tools that (1) yield reliable error control of the finite volume discretization of convection-diffusion-reaction problems in 2-D and 3-D on unstructured grids, and (2) use parallel computational resources efficiently. We achieve balance between obtaining reliable control of the error and efficient use of the available computational resources by an adaptive process of parallel local grid refinement based on a posteriori error analysis.; In our a posteriori error analysis we exploit the ideas known from the finite element method. Namely, we use estimators based on local residuals, local Dirichlet or Neumann problems, and Zienkiewicz-Zhu type estimators. We have constructed such estimators for finite volume approximations and have theoretically justified them by proving, under appropriate assumptions, that they provide both lower and upper bounds for the error. The equivalence of the error and the estimated error is dependent on certain constants. The constants' dependence on the problem's parameters is discussed. The analysis is performed in global H1, L2, and energy norms. Of special interest are problems with large convection. For these problems the standard approximation techniques give oscillating numerical results and mass disbalance. Different up-wind approximations, that give a desired stabilization of the discretization, are discussed and taken into account in the a posteriori error estimators. The results, obtained for steady-state convection-diffusion-reaction problems, are extended to time dependent problems.; The dissertation also carries out extensive numerical testing of the theoretical results. There are two groups of numerical experiments. The first group contains experiments with known exact solutions, for which case the exact error is compared with the numerical results from the error estimators. The second group deals with problems with unknown exact solutions. Also, we have included tests to study the behavior of the error estimators for problems varying from pure diffusion to large convection.; Furthermore, we introduce the parallel local grid refinement algorithms that we implemented into a parallel grid generation tool. Parallel grid generation is essential for the efficient parallel implementation of the adaptive methods and plays an important role in numerical simulations that rely on high performance parallel computers. This part of the dissertation includes element refinement/de-refinement algorithms, techniques to maintain load balance, general issues in developing parallel finite volume (or finite element) code based on domain decomposition, ways to maintain mesh conformity on the different refinement levels, data structures, etc.; In general, we conclude that the ideas from the finite element a posteriori error analysis theory can be successfully used in deriving reliable error estimators for the finite volume method. The efficiency of the error estimators and the parallel grid refinement techniques that we derived is supported by various numerical experiments.