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2.4 Time Dependent Problems
A weak formulation of the problem posed by (2.1), (2.2), and (2.3) is,
![$\displaystyle (v,\mathcal{L}(\mathbf{c}))=(v,\mathbf{f})+\Bigl(v,\frac{\partial \mathbf{c}}{\partial t}\Bigr),\quad \forall v\in \mathcal{V}.$](img70.png) |
(2.15) |
Stating the problem in the space
, we have,
![$\displaystyle (v_h,\mathcal{L}(\mathbf{c}_h))=(v_h,\mathbf{f})+\Bigl(v_h,\frac{\partial \mathbf{c}_h}{\partial t}\Bigr),\quad \forall v_h\in \mathcal{V}_h.$](img71.png) |
(2.16) |
If we take
as a time step and
as an approximation of
at the discrete time
in the space
,
fulfills,
![$\displaystyle (v_h,\mathcal{L}(\mathbf{c}_h^n))=(v_h,\mathbf{f})+\Bigl(v_h,\fra...
...athbf{c}_h^n-\mathbf{c}_h^{n-1}}{\tau}\Bigr),\quad \forall v_h\in\mathcal{V}_h,$](img76.png) |
(2.17) |
The approach using this representation is known as the backward Euler method.
By simple transformation of (2.17) and by introducing the substitution,
![$\displaystyle \mathcal{L}^{'}(\mathbf{c}_h^n)=\tau\mathcal{L}(\mathbf{c}_h^n)-\mathbf{c}_h^n,$](img77.png) |
(2.18) |
![$\displaystyle \mathbf{f}^{'}=\tau\mathbf{f}-\mathbf{c}_h^{n-1},$](img78.png) |
(2.19) |
we obtain,
![$\displaystyle (v_h,\mathcal{L}^{'}(\mathbf{c}_h))=(v_h,\mathbf{f^{'}}),\quad \forall v_h\in \mathcal{V}_h,$](img79.png) |
(2.20) |
which can be treated just like (2.9).
Next: 2.5 Finite Element Spaces
Up: 2. Finite Element Method
Previous: 2.3 Galerkin's Method
H. Ceric: Numerical Techniques in Modern TCAD