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We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. Clifford analogues of complex holomorphic functions—called monogenic functions—are defined by means of the Dirac operators that factor a certain wave operator. One of the fundamental features of quaternionic analysis is the invariance of quaternionic analogues of holomorphic function under conformal (or Möbius) transformations. A similar invariance property is known to hold in the context of Clifford algebras associated to positive definite quadratic forms. We generalize these results to the case of Clifford algebras associated to all non-degenerate quadratic forms. This result puts the indefinite signature case on the same footing as the classical positive definite case. 

In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis-Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.