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I'm currently searching for an elaborate referece that covers most of the approximation errors for elliptic second order problems (like, for the laplacian dirichlet problem) by using finite element methods in 2D and 3D.

For example in 2D, you can receive an optimal error estimate: $$ \| u - R_hu \|_{H^1} = \mathcal{O}(h^r), $$ where $R_h: V \to V_h$ is the Ritz-Projector (aka Galerkin-Projector) from the conformal space $V$, where the variational problem is stated, into the FE-space $V_h$ of piecewise polynomials of degree $r$ (with proper zero conditions on the right boundary edges) on a mesh of mesh-size $h$. Of course, this only works if we assume the solution to be regular enough, i.e. $u \in H^{r+1} \cap V$.

At university, i worked primary with the book of Braess, which is pretty much the standard literature in germany, but it only covers the cases in 2D. I assume similar results can be obtained in 3D, but might get a bit complicated.

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For standard finite elements for elliptic problems we often find no fundamental difference in the n-dimensional analysis (n=2,3,...), since most parts are dimension-agnostic, provided the elements under consideration have optimal approximation properties and we have sufficient regularity (which may be a complicated matter for higher spatial dimensions). Because the regularity of solutions is linked to the type of the domain, some authors tend to fix n=2 and/or certain types of computational domains (e.g. (convex) polygonal) to simplify the presentation. Usually they make a remark that the analysis carries over to more general cases straightforwardly if the regularity is sufficient.

A classical reference covering the n-dimensional case would be e.g. Ciarlet's 1978 book "The Finite Element Method for Elliptic Problems". There are also contributions by Ciarlet and Lions in the "Handbook of Numerical Analysis" series that cover the n-dimensional case.

Sufficient regularity is often not proved in the finite element literature but rather either assumed or referenced (e.g. in Grisvards 1985 book "Elliptic Problems in Nonsmooth Domains" or the book by Gilbarg and Trudinger which Wolfgang mentioned).

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  • $\begingroup$ I read a bit through Ciarlet's "The Finite Element Method for Elliptic Problems". So, for the 3D-case the above statement with the usual $\mathcal{P}_r-Elements$ is exactly the same since $H^{r+1}(\hat{K}) \hookrightarrow C^0(\hat{K})$ for all $r \geq 1$ with the reference tetrahedron $\hat{K}$, right? In four dimensions, this wouldn't hold anymore and we would at least need a solution in $H^3$ instead of the usual $H^2$ by using linear elements. $\endgroup$ – Simul Sep 7 '14 at 14:54
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To prove a convergence rate, there are two steps you need to be able to do: first, you need to be able to show that the Galerkin error can be bounded by a fixed multiple of the interpolation error: $$ \|u-R_h u\|_{H^1} \le C \inf_{\varphi_h \in V_h} \| u-\varphi_h \|_{H^1}. $$ For the Laplace equation, this step is trivial and does not depend on the space dimension. Then, one of course also has $$ \|u-R_h u\|_{H^1} \le C \| u-I_h u \|_{H^{r+1}}. $$

The second step is to show the interpolation estimate $$ \|u-I_h u\|_{H^1} \le C h^{r} \| u \|_{H^{r+1}}. $$ This step, conceptually, only requires something like a Taylor expansion to degree $r$ and consideration of the remainder term. However, it is often shown in a more abstract way where, again, you do not really make use of the space dimension nor that $u$ is actually continuous -- you only need that the solution is in $H^{r+1}$.

So when is $u\in H^{r+1}$? You can find questions like this answered in books like the one by Gilbarg and Trudinger. But to provide the most important case already: if the domain is a convex polyhedron, then $u\in H^2$, in any space dimension.

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