Stable broken \(H^1\) and \(H(\operatorname{div})\) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions. (English) Zbl 1434.65253
Summary: We study extensions of piecewise polynomial data prescribed on faces and possibly in elements of a patch of simplices sharing a vertex. In the \(H^1\) setting, we look for functions whose jumps across the faces are prescribed, whereas in the \(\boldsymbol{H}(\operatorname{div})\) setting, the normal component jumps and the piecewise divergence are prescribed. We show stability in the sense that the minimizers over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken \(H^1\) and \(\boldsymbol{H}(\operatorname{div})\) spaces. Our proofs are constructive and yield constants independent of the polynomial degree. One particular application of these results is in a posteriori error analysis, where the present results justify polynomial-degree-robust efficiency of potential and flux reconstructions.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
polynomial extension operator; broken Sobolev space; potential reconstruction; flux reconstruction; a posteriori error estimate; robustness; polynomial degree; best approximation; patch of elementsReferences:
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