Richard Falk, Rutgers
Abstract: We first provide an introduction to finite element methods, discussing the concepts of consistency, stability, and convergence, and introduce the notion of mixed finite element methods. We next establish a connection to exterior calculus, and in particular to the de Rham complex and to the Hodge Laplacian. We then discuss abstract approximation of the de Rham complex and of boundary value problems for the Hodge Laplacian. The key ideas leading to stable approximation schemes for the Hodge Laplacian are that the approximating spaces form a subcomplex of the de Rham complex and there is a bounded cochain projection from the de Rham complex to the approximating subcomplex. Finally, we outline the construction of two families of finite element spaces of differential forms that satisfy the abstract conditions, and hence are useful in the approximation of a number of important boundary value problems for partial differential equations arising in applications.