  In this paper we analyze a class of trace finite element methods (TraceFEM)
for the discretization of vector-Laplace equations. A key issue in the finite
element discretization of such problems is the treatment of the constraint that
the unknown vector field must be tangential to the surface (``tangent
condition''). We study three different natural techniques for treating the
tangent condition, namely a consistent penalty method, a simpler inconsistent
penalty method and a Lagrange multiplier method. A main goal of the paper is to
present an analysis that reveals important properties of these three different
techniques for treating the tangent constraint. A detailed error analysis is
presented that takes the approximation of both the geometry of the surface and
the solution of the partial differential equation into account. Error bounds in
the energy norm are derived that show how the discretization error depends on
relevant parameters such as the degree of the polynomials used for the
approximation of the solution, the degree of the polynomials used for the
approximation of the level set function that characterizes the surface, the
penalty parameter and the degree of the polynomials used for the approximation
of Lagrange multiplier.
