Error analysis of a space-time finite element method for solving PDEs on evolving surfaces (Q2927847)

The authors analyze the finite element method to obtain approximate solutions of the advection-diffusion equation on moving surfaces NEWLINE\[NEWLINE \dot{u}+ (\operatorname{div}_{\Gamma}w)u -v_{d}\Delta_{\Gamma}u =f, \qquad \text{ on } \quad\Gamma (t)\in \Omega \subset \mathbb R^{d},NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(\cdot,0)=0,\qquad \text{ on }\quad \Gamma_{0}= \Gamma (0), NEWLINE\]NEWLINE with the condition NEWLINE\[NEWLINE \int_{\Gamma (t)}u ds = 0\qquad \text{ for }\quad t\in [0,T].NEWLINE\]NEWLINENEWLINENEWLINEUsing the finite element method based on the regular triangulation of \(\Omega \), it is proved that the convergence of the approximate solutions has the first order in an energy norm and second order in a weaker norm. Numerical examples to illustrate this method are not given.