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Erschienen in:
2021 | OriginalPaper | Buchkapitel
8. Scalar second-order elliptic PDEs
verfasst von : Alexandre Ern, Jean-Luc Guermond
Erschienen in: Finite Elements II
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Abstract
This chapter addresses fundamental properties of scalar-valued second-order elliptic PDEs endowed with a coercivity property. The prototypical example is the Laplacian with homogeneous Dirichlet conditions. More generally, we consider PDEs including lower-order terms, such as the diffusion-advection-reaction equation, where the lower-order terms are assumed to be small enough so as not to pollute the coercivity provided by the diffusion operator. We also study in some detail how various boundary conditions (Dirichlet, Neumann, Robin) can be enforced in the weak formulation. Moreover, important smoothness properties of the solutions to scalar second-order elliptic PDEs are listed at the end of the chapter.