Geodesic
Nitsche’s method for parabolic partial differential equations with mixed time varying boundary conditions
Tagliabue, Anna1 ; Dedè, Luca2  ; Quarteroni, Alfio1, 2 
1 MOX – Modeling and Scientific Computing, Mathematics Department “F. Brioschi”, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy.
2 CMCS – Chair of Modeling and Scientific Computing, MATHICSE – Mathematics Institute of Computational Science and Engineering, EPFL – École Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 541-563

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We investigate a finite element approximation of an initial boundary value problem associated with parabolic Partial Differential Equations endowed with mixed time varying boundary conditions, switching from essential to natural and vice versa. The switching occurs both in time and in different portions of the boundary. For this problem, we apply and extend the Nitsche’s method presented in [Juntunen and Stenberg, Math. Comput. (2009)] to the case of mixed time varying boundary conditions. After proving existence and numerical stability of the full discrete numerical solution obtained by using the θ-method for time discretization, we present and discuss a numerical test that compares our method to a standard approach based on remeshing and projection procedures.

DOI : 10.1051/m2an/2015054
Classification : 35K20, 65M12, 65M60, 68U20, 74S05
Keywords: Nitsche’s method, parabolic problems, mixed time varying boundary conditions, stability analysis, finite element method

Tagliabue, Anna 1 ; Dedè, Luca 2 ; Quarteroni, Alfio 1, 2

1 MOX – Modeling and Scientific Computing, Mathematics Department “F. Brioschi”, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy.
2 CMCS – Chair of Modeling and Scientific Computing, MATHICSE – Mathematics Institute of Computational Science and Engineering, EPFL – École Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland 
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     title = {Nitsche{\textquoteright}s method for parabolic partial differential equations with mixed time varying boundary conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {541--563},
     publisher = {EDP-Sciences},
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Tagliabue, Anna; Dedè, Luca; Quarteroni, Alfio. Nitsche’s method for parabolic partial differential equations with mixed time varying boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 541-563. doi: 10.1051/m2an/2015054

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