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linear parabolic equation; third boundary condition; finite element method; semidiscretization; fully discretized scheme; elliptic projection
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.
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