Summary Mortar methods with dual Lagrange multiplier bases provide a flexible, efficient and optimal way to couple different discretization schemes or nonmatching triangulations. Here, we generalize the concept of dual Lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the Lagrange multiplier space have the same dimension. We provide a new theoretical framework within this relaxed setting, which opens a new and simpler way to construct dual Lagrange multiplier bases for higher order finite element spaces. As examples, we consider quadratic and cubic tetrahedral elements and quadratic serendipity hexahedral elements. Numerical results illustrate the performance of our approach.

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Abstract. We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. 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. Working with Gauß-Lobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are two-dimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.