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I would like to know if there are any similar theorems related to the mass lumping finite element method:

For any function $\boldsymbol{u} \in H^m_{per}(\Omega)$ $$ \|I_h e^{\tau \Delta} \boldsymbol{u} - e^{\tau \Delta_h} I_h \boldsymbol{u}\| \leq C h^m |u|_m. $$ where $I_h$ is the grid interpolation operator, $\Delta$ is the Laplace operator, and $\Delta_h$ is its discrete form (in particular, $e^{\tau \Delta_h}$ is an exponential matrix). Here, $h$ is the grid size. The above theorem can be easily applied to the finite difference method or the spectral method.

I would like to know if there are any relevant papers that provide proofs for finite element functions. It would be great if you could provide links to the papers. Thank you in advance!

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    $\begingroup$ Can you be more specific what it is you want? Do you mean that in the finite element context, you want $I_h$ to interpolate not onto a set of points, but instead into a finite-dimensional sub-space of finite element functions? $\endgroup$
    – Wolfgang Bangerth
    Commented Sep 13 at 21:29
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    $\begingroup$ Are you actually computing those exponentials in your implementation ? The notation $e^{\tau\Delta}u$ just means it is the exact solution of your PDE. And $e^{\tau\Delta_h} I_h u$ is the exact solution of your FEM assuming time integration is exact. And the inequality you wrote is just the error estimate of your FEM under exact time integration, which is what you would usually prove as a first step of error analysis. $\endgroup$
    – cfdlab
    Commented Sep 15 at 3:25
  • $\begingroup$ @WolfgangBangerth Yes, I have conducted some numerical experiments, and for basis functions of order $k$, the finite element method based on the Legendre–Gauss–Lobatto approach can achieve $k+1$-order convergence. This means I need to obtain an estimate similar to $\leq C h^{k+1}$ . The above theorem is derived based on spectral methods (Lemma 3.4) [doi.org/10.1137/15M1041122], but how should this be applied to the finite element method with Legendre–Gauss–Lobatto? I would like to learn some proof techniques in this regard. $\endgroup$
    – Owen Jun
    Commented Sep 20 at 1:43
  • $\begingroup$ @cfdlab I would like to know whether the accuracy of the time integration implies that finite element interpolation error estimates can be applied directly. For example, can we directly obtain $\|I_h e^{\tau \Delta} u - e^{\tau \Delta_h} I_h u\|_2 \leq Ch^{k+1}$, where k is the highest order of the basis functions? $\endgroup$
    – Owen Jun
    Commented Sep 20 at 1:47

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