UC San Diego

In this dissertation, we formulate and implement p- adaptive and hp-adaptive finite element methods to solve elliptic partial differential equations. The main idea of the work is to use elements of high degrees solely (p- adaptive) or in combination with elements of small size (hp-adaptive) to better capture the behavior of the solution. In implementing the idea, we deal with different aspects of building an adaptive finite element method, such as defining basis functions, developing algorithms for adaptive meshing procedure and formulating a posteriori error estimates and error indicators. The basis functions used in this work are regular nodal basis functions and special basis functions defined for elements with one or more edges of higher degree transition elements). It is proved that with our construction of these basis functions, the finite element space is well- defined and C⁰. Several algorithms are developed for different scenarios of the adaptive meshing procedure, namely, p-refinement, p-unrefinement and hp-refinement. They all follow the 1-irregular rule and 2-neighbor rule motivated by [Bank and Sherman, 1983 - MR751598]. These rules help to limit the number of special cases and maintain the sparsity of the stiffness matrix, and thus to simplify the implementation and reduce the cost of calculation. The work of formulating a posteriori error estimates and error indicators is the core of this dissertation. Our error estimates and error indicators are based on the derivative recovery technique proposed by [Bank and Xu, 2003 - MR2034616, MR2034617] and Bank et al., 2007 - MR2346369]. Using the information in formulating the error indicators, we define a hp-refinement indicator which can be utilized to decide whether a given element should be refined in h or in p. Numerical results show that the combination of the two indicators helps automatic hp-refinement to create optimal meshes that demonstrate exponential rate of convergence. In this dissertation, we also consider hp-adaptive and domain decomposition when they are combined using the parallel adaptive meshing paradigm developed by [Bank and Holst, 2000 - MR1797889]. Numerical experiments demonstrate that the paradigm scales up to at least 256 processors (maximum size of our experiments) and with nearly 200 millions degrees of freedom