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This book discusses a family of computational methods, known as discontinuous Galerkin methods, for solving partial differential equations. While these methods have been known since the early 1970s, they have experienced an almost explosive growth interest during the last ten to fifteen years, leading both to substantial theoretical developments and the application of these methods to a broad range of problems. These methods are different in nature from standard methods such as finite element or finite difference methods, often presenting a challenge in the transition from theoretical developments to actual implementations and applications.