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"This book offers on the one hand a complete theory of Sobolev spaces, which are of fundamental importance for elliptic linear and non-linear differential equations, and explains on the other hand how the abstract methods of convex analysis can be combined with this theory to produce existence results for the solutions of non-linear elliptic boundary problems. The book also considers other kinds of functional spaces which are useful for treating variational problems such as the minimal surface problem."

"Partial differential equations (PDEs) are essential for modeling many physical phenomena. This undergraduate textbook introduces students to the topic with a unique approach that emphasizes the modern finite element method alongside the classical method of Fourier analysis.Additional features of this new edition include broader coverage of PDE methods and applications, with new chapters on the method of characteristics, Sturmiouville problems, and Green functions, and a new section on the finite difference method for the wave equation. The author continues to emphasize Fourier series and finite element methods, which were the primary scope of the first edition."

"This introductory text on partial differential equations is the first to integrate modern and classical techniques for solving PDEs at a level suitable for undergraduates. The author successfully complements the classical topic of Fourier series with modern finite element methods. The result is an up-to-date, powerful, and flexible approach to solving PDEs, which both faculty and students will find refreshing, challenging, and rewarding. Linear algebra is a key component of the text, providing a framework both for computing solutions and for understanding the theoretical basis of the methods. Although techniques are emphasized over theory, the methods are presented in a mathematically sound fashion to develop a strong foundation for further study. Numerous exercises and examples involve meaningful experiments with realistic physical parameters, allowing students to use physical intuition to understand the qualitative features of the solutions."

"A concise survey of the current state of knowledge in 1972 about solving elliptic boundary-value eigenvalue problems with the help of a computer. This volume provides a case study in scientific computing -- the art of utilizing physical intuition, mathematical theorems and algorithms, and modern computer technology to construct and explore realistic models of problems arising in the natural sciences and engineering."

"ABSTRACTThis paper will show a comparison between the Kepler, Maxwell and Pascal GPUarchitectures using CUDA-Fortran, with and without dynamic calls, to efficiently solve partialdifferential equations. The target is to show the possibility of using affordable hardware, such asthe GTX670, GTX970 and GTX1080 NVIDIA GPUs, which are commonly found in personaland portable computers, for scientific applications. For simplicity we consider a standardwave equation where we use a second order finite difference method for the spatial and timediscretizations to obtain the numerical solution. We found that, as we increase the spatialresolution of the domain we also increase the performance difference between the GPU and theCentral Processing Unit (CPU)"

"Intended mainly for use as a reference manual, this third edition encompasses all the improvements of the newest version of the PLTMG software package. This updated version introduces several significant changes, including the following:The data structures defining the domain (triangulation and skeleton) have been simplified; more general boundary conditions are allowed.The internal data structure (triangle tree) has been replaced by a simple triangulation; the adaptive algorithms have been revised to use this simple data structure.The linear equations solvers are multigraph methods, which are simple ILU factorizations with strong connections to the hierarchical basis multigrid method.The graphics packages have been improved and now include expanded options.The test driver has a new X-Windows gui option."