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A trace finite element method for vector-Laplacians on surfaces. (English) Zbl 1402.65158

The authors are concerned with a specialized finite element method for the discretization of a vector-Laplace problem posed on a two-dimensional surface embedded in a three-dimensional domain. They introduce some notions of tangential differential calculus, give a weak formulation of the problem with a Lagrange multiplier, and show that it is well posed. A trace FEM is formulated along with a discrete LBB stability condition for certain pairs of trace finite element spaces. Then an optimal order error estimate in the energy norm is provided. Some algebraic properties of the resulting saddle point stiffness matrix including an optimal Schur complement preconditioner are studied. Two vector-Laplace problems, one on the unit sphere and another on a planar surface, are solved in order to illustrate the capabilities of the method.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65D05 Numerical interpolation
65N15 Error bounds for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
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References:

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