On the L2 Stability of the 1-d Mortar Projection

It is previously known that the one dimensional mortar nite element projection is stable in the L 2 norm, provided that the ratio of any two neighboring mesh intervals is uniformly bounded, but with the constant in the bound depending on the maximum value of that ratio. In this paper, we show that this projection is stable in the L 2 norm, independently of the properties of the nonmortar mesh. The 1D trace of the mortar space considered here is a piecewise polynomial space of arbitrary degree; therefore, our result can be used for both the h and the hp version of the mortar nite element. 1. Introduction. Mortar nite elements are nonconforming nite elements that allow for nonconforming decomposition of the computational domain and for the optimal coupling of diierent variational approximations in diierent subregions. Here, by optimality we mean that the global error is bounded by the sum of the local best approximation errors on each subregion. Because of these features, the mortar elements are quite general and they are used eeectively in solving large classes of problems. The mortar nite element methods were rst introduced by Bernardi, Maday, and Patera in 4]. A three dimensional version was developed by Ben Belgacem and Maday in 3], and was further analyzed for three dimensional spectral elements in 2]. Let us brieey describe the mortar nite element space V h underlying the mortar method. The computational domain is decomposed into a nonoverlapping polygonal partition f k g k=1:K. Since we are working with geometrically nonconforming mortars, we do not require that the intersection of the boundaries of two diierent subregions be either empty, or a vertex, or an entire edge. The restriction of the mortar space to any subregion i is a conforming P m i or Q m i nite element space. In other words, i is partitioned in a geometrically conforming fashion into triangles or quadrilaterals, and the restriction of V h to each element of this partition is a polynomial of total degree m i (for P m i), or of degree m i in each variable (for Q m i). Since our arguments will be local, the degrees m i are completely arbitrary. Across the interface ?, i.e. the set of points that belong to the boundaries of at least two subregions, pointwise continuity is not required. We partition ? into a union of nonoverlapping edges …