Chapter 7 focuses upon response of continuous systems. The shear beam is used as the prime example. As a previous chapters, deterministic vibration via modal solutions initiates the subject. Stationary and concentrated random excitation follow in order. The author touches upon random vibration of a thin plate. Our next topic is the alternative solution of shear beams, where again, the uncoupled SDOF systems are applied in either time or frequency domain. The random vibration of a dam reservoir system subjected to vertical ground acceleration is exploited. The alternative impulse response function solves the dam problem where the hydrodynamic pressure cor­ responds to the shear force. The frequency response function and power spectral density response conclude this section. In a similar manner, the dam-reservior is excited by a horizontal acceleration simulating an earthquake event. A very good chapter! The next chapter covers the design of structures for random excitations. The stationary Gaussian process (random variable) is defined. The probability of upcrossings over a specified level and probability density of the peaks introduce this phase of the subject. The author applies this to the con­ cept of structural design against yield failure using an SDOF and a dam-reservoir subjected to vertical ground acceleration. The chapter concludes with structural damage against fatigue and includes the Palmgren-Miner (PM) rule and stationary narrow band random loading. The reviewer would have preferred seeing actual random test data since PM is limited as to excitations and may be unconservativ e at times. Chapter 9 treats nonstationary response which is a time dependent response of structures. Beginning with an SDOF system having stationary excitation, we again utilize the damreservoir system (vertical acceleration excitation). This con­ tinues into systems with nonstationary excitation containing Priestley's complex integral model. Zero damped and lightly damped SDOF systems inaugurate this topic. An alternative approach is Bendat and Piersol's model which employs the autocorrelation function for zero mean excitation and nonsta­ tionary specturm. The dam-reservoir with nonstationary ex­ citation concludes the chapter. The last chapter develops random vibration of structures in the plastic range. The random walk model is the eye-opener. It encompasses basic probability definitions (mass probability, mass function, conditional probability) and then derives the Chapman-Kalmagarov-Smolchoewski equation applied to structural response. Application of random walk model to a nonlinear structure and solution of the Fokker-Plunck equa­ tion complete the chapter. The appendices include Fast Fourier Transform (FFT) in random vibration and Monte Carlo simulation plus references. In summary, this is a good book. The author spares no ef­ fort in making the reader understand random vibration. The

We present in this paper a new a posteriori error estimator for the Baumann-Oden version of the Discontinuous Galerkin Method. The error estimator is based on the residual of the partial differential equation. In the case of the reaction-diffusion equation, the norm of the residual is shown to be equivalent to the error in some specific energy-type norms. We propose here a method to efficiently calculate the norm of the residual and present some numerical experiments which demonstrate the reliability of the methodology. 1 Introd uction In the present paper, we report on new results dealing with a posteriori error estimation for hp approximations obtained by a slightly modified (stabilized) version of the Baumann-Oden Discontinuous Galerkin finite element method [1, 2]. For theoretical purposes, we consider here the Poisson problem as the model problem. To the best knowledge of the authors, there is to date no published work on this subject, except maybe the experimental work of Riviere et at. [4, 5], which introduces an a posteriori error estimator for the same class of problems, but lacks a complete theoretical basis. We also show some preliminary numerical results obtained on a two-point boundary value problem to demonstrate the effectivity of our methodology. The paper is organized as follows. Section 2 introduces some notations as well as the model problem and the discontinuous Galerkin formulation. In Section 3, we present the theoretical basis to derive the a posteriori error estimator. Numerical experiments are described in Section 4, and some concluding remarks are given in Section 5. 2 Notations and preliminaries Let 0 denote a bounded open set in lR d , d = 1 or 2, with Lipschitz continuous boundaryaO. The parts of the boundary on which Dirichlet and Neumann conditions are prescribed are respectively denoted rD and rN, such that rD U fN = ao, and rVnrN = 0. A finite element partition Ph of 0 consists of a finite collection of Ne open elements Introducing the size hK of an clement K as the diameter of K, we associate with each partition the parameter h: h = maxKEPhhK. In addition, the boundary of an element K is denoted 8K, while the unit normal vector outward from K is denoted by n. Given a partition Ph, the collection of element interfaces (points, edges, and faces, in one, two, and three dimensions respectively) is denoted by the set E h = …