Geodesic
Galerkin approximations for the linear parabolic equation with the third boundary condition
Applications of Mathematics, Tome 48 (2003) no. 2, pp. 111-128.

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We solve a linear parabolic equation in $\mathbb{R}^d$, $d \ge 1,$ with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the $\theta $-method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.
DOI : 10.1023/A:1026042110602
Classification : 65M12, 65M15, 65M60
Keywords: linear parabolic equation; third boundary condition; finite element method; semidiscretization; fully discretized scheme; elliptic projection 
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     title = {Galerkin approximations for the linear parabolic equation with the third boundary condition},
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Faragó, István; Korotov, Sergey; Neittaanmäki, Pekka. Galerkin approximations for the linear parabolic equation with the third boundary condition. Applications of Mathematics, Tome 48 (2003) no. 2, pp. 111-128. doi : 10.1023/A:1026042110602. http://geodesic.mathdoc.fr/articles/10.1023/A:1026042110602/

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