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The finite element method is a popular method for approximating numerically on a computer the solution of partial differential equations. It is based on a variational (weak) formulation of the PDE, followed by discretization on a finite-dimensional ansatz space which reduces the problem to a sparse linear algebra one.

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stability of parabolic problems where nonhomogeneous term in $L^2(0,T; H^{-1}(\Omega))$

There is a problem in this estimate. In Part 2, We can't derive the second inequality from the first inequality in Part 2. The main reason is the last term on the RHS in the first inequality is in $\| …