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The particular solutions of the Laplace equations and their singularities are fundamental to numerical partial di erential equations in both algorithms and error analysis. We first review the explicit solutions of Laplace¡¦s equations on sectors with the Dirichlet and the Neumann boundary conditions. These harmonic functions clearly expose the solution¡¦s regularity/singularity at the vertex. So we can analyze the singularity of the Laplace¡¦s solutions on polygons at di erent domain corners and for various boundary conditions. By using this knowledge we can designed many new testing models with di erent kind of singularities, like discontinuous and mild singularities, beside the popular singularity models, Motz¡¦s and the cracked beam problems, We use the boundary approximation method, i.e. the collocation Tre tz method in engineering literatures, to solve the above testing models of Laplace boundary value problems on polygons. Suppose the uniform particular solutions are chosen in the entire domain. When there is no singularity on all corners, this method has the exponential convergence. However, its rate of convergence will deteriorate to polynomial if there exist some corner singularities. From experimental data, we even have three type of convergence, i.e. exponential, polynomial or their mixed types. We will study these convergent behaviors and their causes. Finally, we will uncover the relation between the order of convergence and the intensity of corner singularities.

This dissertation presents several solutions on the simulation of weakly nonlinear circuits. The work is motivated by the increasing demand on fast yet accurate simulation methods circuits (IC)s, and the current lack of such methods in the electronic design automation (EDA) / computer-aided design (CAD) community. Three types of frequency domain methods are studied to analyze weakly nonlinear circuits. The first method employs numerical multi-dimensional inverse Laplace transform based on Laguerre function expansion. An adaptive mesh refinement (AMR) technique is developed and its parallel implementation is introduced to speed up the computation. The second method applies a Fourier series based algorithm to invert Laplace transform. The algorithm is straightforward to implement, and gives increasing accuracy with increasing number of frequency sampling points. It employs a fast Fourier transform (FFT)-based method to directly invert the frequency domain solution. Its parallel routine is also studied. The third method is based on Gaver functional. It enjoys a high accuracy independent of the number of sampling points, and for multidimensional simulation, only the diagonal points in the matrix are required to be computer, which can be further speeded up by parallel implementation. Numerical results show that the aforementioned three methods enjoy good accuracy as well as high efficiency. A comparative study is carried out to investigate the strengths and drawbacks of each method. / published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy

Thesis (Ph. D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 1993. / Includes bibliographical references (leaves 272-277).

Thesis--Technical University of Denmark. / With Danish resumé. Bibliography: p. [75]-76.

Thesis--Technical University of Denmark. / With Danish resumé. Bibliography: p. [75]-76.

Thesis (Ph. D.)--Oregon State University, 2004. / Typescript (photocopy). Includes bibliographical references (leaves 123-125). Also available via the World Wide Web.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Universiẗat, Diss., 2001--Mainz.