On the Poincaré-Friedrichs inequality for piecewise \(H^{1}\) functions in anisotropic discontinuous Galerkin finite element methods. (English) Zbl 1208.26023
Summary: The purpose of this paper is to propose a proof for the Poincaré-Friedrichs inequality for piecewise \( H^1\) functions on anisotropic meshes. By verifying suitable assumptions involved in the newly proposed proof, we show that the Poincaré-Friedrichs inequality for piecewise \( H^1\) functions holds independently of the aspect ratio which characterizes the shape-regular condition in finite element analysis. In addition, under the maximum angle condition, we establish the Poincaré-Friedrichs inequality for the Crouzeix-Raviart nonconforming linear finite element. Counterexamples show that the maximum angle condition is only sufficient.
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
shape-regular condition; anisotropic mesh; Crouzeix-Raviart nonconforming linear element; the maximum angle conditionReferences:
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