| Abstract: | Suppose $cal{S}^1({cal T})subset H^1(Omega)$ is the $P_1$-finite elementspace of $cal{T}$-piecewise affine functions based on a regular triangulation $cal{T}$ of a two-dimensional surface$Omega$ into triangles.The $L^2$ projection $Pi$ onto $cal{S}^1(cal{T})$ is $H^1$ stableif $norm{Pi v}{H^1(Omega)}le Cnorm{v}{H^1(Omega)}$ forall $v$ in the Sobolev space $H^1(Omega)$ and if the bound $C$does not depend on the mesh-size in $cal{T}$ or on thedimension of $cal{S}^1(cal{T})$.hskip 1em A red–green–blue refining adaptive algorithm is designed whichrefines a coarse mesh $cal{T}_0$ successively such that each triangle isdivided into one, two, three, or four subtriangles. This is the newest vertex bisection supplemented with possible red refinementsbased on a careful initialization.The resulting finite element space allows for an $H^1$ stable $L^2$ projection.The stabilitybound $C$ depends only on the coarse mesh $cal{T}_0$ through the number ofunknowns, the shapes of the triangles in $cal{T}_0$, and possibleDirichlet boundary conditions. Our arguments alsoprovide a discrete version $norm{h_cal{T}^{-1},Pi v}{L^2(Omega)}le Cnorm{h_cal{T}^{-1},v}{L^2(Omega)}$in $L^2$ norms weighted with the mesh-size $h_T$. |
