The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.

This book provides insight into the mathematics of Galerkin finite element method as applied to parabolic equations. The revised second edition has been influenced by recent progress in application of semigroup theory to stability and error analysis, particulatly in maximum-norm. Two new chapters have also been added, dealing with problems in polygonal, particularly noncovex, spatial domains, and with time discretization based on using Laplace transformation and quadrature.

This book provides insight into the mathematics of Galerkin finite element method as applied to parabolic equations. The revised second edition has been influenced by recent progress in application of semigroup theory to stability and error analysis, particulatly in maximum-norm. Two new chapters have also been added, dealing with problems in polygonal, particularly noncovex, spatial domains, and with time discretization based on using Laplace transformation and quadrature.

The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.

Das Buch ist fur Studenten der angewandten Mathematik und der Ingenieurwissenschaften auf Vordiplomniveau geeignet. Der Schwerpunkt liegt auf der Verbindung der Theorie linearer partieller Differentialgleichungen mit der Theorie finiter Differenzenverfahren und der Theorie der Methoden finiter Elemente. Fur jede Klasse partieller Differentialgleichungen, d.h. elliptische, parabolische und hyperbolische, enthalt der Text jeweils ein Kapitel zur mathematischen Theorie der Differentialgleichung gefolgt von einem Kapitel zu finiten Differenzenverfahren sowie einem zu Methoden der finiten Elemente. Den Kapiteln zu elliptischen Gleichungen geht ein Kapitel zum Zweipunkt-Randwertproblem fur gewohnliche Differentialgleichungen voran. Ebenso ist den Kapiteln zu zeitabhangigen Problemen ein Kapitel zum Anfangswertproblem fur gewohnliche Differentialgleichungen vorangestellt. Zudem gibt es ein Kapitel zum elliptischen Eigenwertproblem und zur Entwicklung nach Eigenfunktionen. Die Darstellung setzt keine tiefer gehenden Kenntnisse in Analysis und Funktionalanalysis voraus. Das erforderliche Grundwissen uber lineare Funktionalanalysis und Sobolev-Raume wird im Anhang im Uberblick besprochen. "