1 - 9 of 9 Chapters
[The background and motivation for the development of solution methodologies for partial differential equations are given with an overview of the related work and the relevant publications.]
[Discretization principles as a foundation for numerical approaches and PDE solution methodologies are described, for space and time, from a bird’s-eye view. Two PDE solution methodologies, based on the strong or the weak form, in mesh-based and meshless methods, are introduced briefly.]
[Essential supporting algorithms for discretization of the domain, finding the nearest nodes, numerical integration, MLS approximation or interpolation, and solution of linear system of equations, which are all needed in the implementation of PDE solution methodologies, are discussed and...
[The construction of the global system of equations, which is one of the main steps of the PDE solution process, is described for the mesh-based methods FDM and FEM, which are a strong and a weak form method, respectively. Its asymptotic calculation complexities are evaluated.]
[Strong form MLSM and weak form MLPG1 meshless methods are presented. The methodology is presented in terms of construction of the global system. The asymptotic calculation complexities are evaluated for both methods.]
[Solution procedures of four methods, FDM, FEM, MLSM, and MLPG1, are presented on a diffusion equation. All the methods are assessed through experimental results for accuracy and execution time.]
[The meshless solutions of test cases are shown on mechanics of a cantilever beam, on fluid flow simulation, and on the simulation of semiconductor PN junction.]
[The described algorithms and methodologies are analyzed regarding their suitability for parallel execution on multiple processors that act as multicore processors, GPUs, or distributed computers. The speedup and runtime have been assessed experimentally.]
[A summary of the presented methods and their comparison is provided. A short history of computing developments and their impact on the development of numerical analysis is presented. Some points relating to implementations on modern computers are raised, also in the light of future developments.]
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