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An extension theorem for polynomials on triangles. (English) Zbl 1175.65143

Summary: We present an extension theorem for polynomial functions that proves a quasi-optimal bound for a lifting from \(L^{2}\) on edges onto a fractional-order Sobolev space on triangles. The extension is such that it can be further extended continuously by zero within the trace space of \(H^{1}\). Such an extension result is critical for the analysis of non-overlapping domain decomposition techniques applied to the \(p\)-and \(hp\)-versions of the finite and boundary element methods for elliptic problems of second order in three dimensions.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J25 Boundary value problems for second-order elliptic equations
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