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"Discontinuous Galerkin (DG) methods for solving partial differential equations, developed in the late 1990s, have become popular among computational scientists. This book covers both theory and computation as it focuses on three primal DG methods -- the symmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and nonsymmetric interior penalty Galerkin which are variations of interior penalty methods. The author provides the basic tools for analysis and discusses coding issues, including data structure, construction of local matrices, and assembling of the global matrix. Computational examples and applications to important engineering problems are also included."

"This book introduces the basic ideas to build discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite element and finite volume viewpoints are exploited to convey the main ideas underlying the design of the approximation. "

"The issue of regularity has played a central role in the theory of Partial Differential Equations almost since its inception, and despite the tremendous advances made it still remains a very fruitful research field. The course aimed to show the deep connections between these topics and to open new research directions through the contributions of leading experts in all of these fields."

"Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods. A language loaded with acronyms for a thousand different algorithms has developed, and it is often difficult even for specialists to identify the basic principles involved. Here is a book that focuses on the analysis of iterative methods. The author includes the most useful algorithms from a practical point of view and discusses the mathematical principles behind their derivation and analysis. Several questions are emphasized throughout: Does the method converge? If so, how fast? Is it optimal, among a certain class? If not, can it be shown to be near-optimal? The answers are presented clearly, when they are known, and remaining important open questions are laid out for further study. Greenbaum includes important material on the effect of rounding errors on iterative methods that has not appeared in other books on this subject. Additional important topics include a discussion of the open problem of finding a provably near-optimal short recurrence for non-Hermitian linear systems; the relation of matrix properties such as the field of values and the pseudospectrum to the convergence rate of iterative methods; comparison theorems for preconditioners and discussion of optimal preconditioners of specified forms; introductory material on the analysis of incomplete Cholesky, multigrid, and domain decomposition preconditioners, using the diffusion equation and the neutron transport equation as example problems. A small set of recommended algorithms and implementations is included."

"This second edition provides much-needed updates to the original volume. Like the first edition, it emphasizes the ideas behind the algorithms as well as their theoretical foundations and properties, rather than focusing strictly on computational details; at the same time, this new version is now largely self-contained and includes essential proofs.Additions have been made to almost every chapter, including an introduction to the theory of inexact Newton methods, a basic theory of continuation methods in the setting of differentiable manifolds, and an expanded discussion of minimization methods. New information on parametrized equations and continuation incorporates research since the first edition."

"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."

"ABSTRAKTujuan utama dari keamanan komunikasi adalah mencegah data pesan dari akses pihak ketiga. Solusi dari masalah tersebut adalah menerapkan enkripsi End-to-end. Algoritma enkripsi Elliptic Curve Cryptography (ECC) cocok digunakan sebagai metode enkripsi End-to-end pada sistem komunikasi digital termasuk aplikasi chat. ECC adalah jenis kriptografi kunci publik yang mendasarkan keamanannya pada permasalahan matematis dari kurva eliptik, ECC mempunyai tingkat keamanan yang setara dengan RSA menggunakan kunci berukuran lebih kecil. Penelitian ini membahas perancangan dan implementasi dari algoritma enkripsi dan dekripsi yang menerapkan kriptografi ECC sebagai metode enkripsi End-to-end pada aplikasi chat. Algoritma ini menggunakan operasi matematika dan nilai parameter dari kurva eliptik untuk menghasilkan kunci serta ciphertextnya dan juga melakukan dekripsi. Dari hasil ujicoba dan analisa implementasi enkripsi dan dekripsi menggunakan algoritma ECC, algoritma tersebut cocok digunakan sebagai metode enkripsi End-to-end, karena ukuran kunci yang relatif ringan dan operasi matematika yang menggunakan parameter khusus yang berbeda-beda. Hasil pengujian enkripsi dan dekripsi menggunakan parameter kurva eliptik secp192r1 dan secp192k1 yang mempunyai ukuran kunci sekitar 2¹⁹² bit, menghasilkan kunci dan ciphertext paling ringan dan paling cepat diproses, dengan angka nilai keduanya berukuran sepanjang 58 digit.

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ABSTRACTThe main purpose of communication security is to prevent message data from third party access. The solution to this problem is to implement End-to-end encryption. Elliptic Curve Cryptography (ECC) encryption algorithm is suitable for use as an End-to-end encryption method on digital communication systems including chat applications. ECC is a type of public key cryptography that bases its security on mathematical problems from elliptic curves, ECC has a level of security equals to RSA while using smaller keys. This study discusses the design and implementation of encryption and decryption algorithms that apply ECC cryptography as an End-to-end encryption method in chat applications. This algorithm uses mathematical operations and parameter values from the elliptic curve to generate keys and their ciphertext and also decrypt them. From the results of testing and analyzing the implementation of encryption and decryption using the ECC algorithm, the algorithm is suitable for use as an End-to-end encryption method, because of the relatively light key size and mathematical operations that use different parameters. The results of encryption and decryption testing using elliptic curve parameters secp192r1 and secp192k1 which have a key size of about 2¹⁹² bits, produce the lightest and fastest to process ciphertext and keys, with the value of both measuring 58 digits.

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