High Order 2D Finite Element Methods with Extra SmoothnessdissertationsParker, Charles WilliamcreatorAinsworth, MarkAdvisorSüli, EndreReaderKarniadakis, GeorgeReaderBrown University. Department of Applied Mathematicssponsor2022xiii, 298 p.born digitalThesis (Ph. D.)--Brown University, 2022thesesFinite element methods (FEMs) are used to discretize partial differential equations in a wide array of applications, including fluid flow, solid mechanics, electromagnetism, optimal control, and magnetohydrodynamics. While much of the theory and implementation of high order continuous FEMs are well-understood, FEMs with more smoothness are less popular in the literature and in practice. In particular, engineering problems involving plates and shells often give rise to fourth order elliptic equations, whose conforming approximations often entail elements with additional smoothness. Methods with extra smoothness also arise in mass-conserving discretizations of incompressible flow problems. This thesis considers theoretical and practical issues related to using these methods. In the first part of this thesis, we construct the first non-overlapping Additive Schwarz Method preconditioner for high order finite elements with C1 continuity on unstructured meshes of triangles. We show that the condition number of the preconditioned system is bounded independently of the mesh size and grows slowly in the polynomial degree. In the second part of this thesis, we discuss the discretization of Stokes flow by a pointwise divergence-free FEM with extra smoothness at element vertices. We present uniform (in mesh size and polynomial degree) stability results for this method. Additionally, the method possesses optimal convergence properties for both the velocity and pressure variables. We then address the issue of preconditioning the underlying saddle point problem with a block diagonal preconditioner.Numerical analysispreconditionersFinite element methodEnglishRPB20220706