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Biorthogonal bases with local support and approximation properties. (English) Zbl 1121.65120

Locally supported basis functions, which are biorthogonal to the conforming nodal finite element basis functions of degree $p$ in one dimension, are constructed. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree $p-1$.

Various theoretical properties of such bases are derived with proofs. In particular, if the Gauss-Lobatto nodes are used, an interesting connection between biorthogonality and quadrature formulas holds. One important application of these newly constructed biorthogonal bases are two-dimensional mortar finite elements based on a new dual Lagrange multiplier space. The weak continuity condition of the constrained mortar space is realized in terms of the new dual bases. As a result, local static condensation can be applied, which is very attractive from the numerical viewpoint.

Numerical results are presented for cubic mortar finite elements, and the discretization errors with convergence rates are measured in various norms. However, a theoretical error analysis is not performed here, so that comparison of the observed convergence rates with the theoretically predicted ones can be an interesting subject.

##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N55 Multigrid methods; domain decomposition (BVP of PDE) 65N12 Stability and convergence of numerical methods (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems