Engineering Notes on Concepts of the Finite Element Method for Elliptic Problems

Abstract

In this contribution, we discuss some basic mechanical and mathematical features of the finite element technology for elliptic boundary value problems. Originating from an engineering perspective, we will introduce step by step of some basic mathematical concepts in order to set a basis for a deeper discussion of the rigorous mathematical approaches. In this context, we consider the boundedness of functions, the classification of the smoothness of functions, classical and mixed variational formulations as well as the $H^{-1}$-FEM in linear elasticity. Another focus is on the analysis of saddle point problems occurring in several mixed finite element formulations, especially on the solvability and stability of the associated discretized versions.

A Sobolev and Hilbert Spaces

In the following we will use the Sobolev and Hilbert Spaces, they are based on the space of square integrable functions on ${\mathcal B}$:

$$\begin{aligned} L^2 ({\mathcal B}) = \big \{ u : \Vert u\Vert ^2_{L^2 ({\mathcal B})} = \int _{{\mathcal B}} |u|^2 \text {d}v < + \infty \big \} \, . \end{aligned}$$

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Let $s \ge 0$ be a real number, the standard notation for a Sobolev space is $H^s ({\mathcal B})$ and $H^s (\partial {\mathcal B})$ with the inner products and norm

$$\begin{aligned} (u, u)_{s, {\mathcal B}} \, , \quad (u, u)_{s, \partial {\mathcal B}} \quad \text {and}\quad \Vert u\Vert _{s, {\mathcal B}} \, , \quad \Vert u\Vert _{s, \partial {\mathcal B}} \, , \end{aligned}$$

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respectively. For $s = 0$ the space $H^0 ({\mathcal B})$ represents the Hilbert space $L^2 ({\mathcal B})$ of all square integrable functions, i.e.,

$$\begin{aligned} L^2 ({\mathcal B}) = H^0 ({\mathcal B}) = \{ u \in L^2 ({\mathcal B}) \} \, . \end{aligned}$$

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If s is a positive integer the spaces $H^s ({\mathcal B})$ consist of all square integrable functions whose derivatives up to the order s are also square integrable, i.e.,

$$\begin{aligned} H^s ({\mathcal B}) = \big \{ u + \sum _{\alpha = 1}^{s} {\text {D}}^{\alpha } u \in L^2 ({\mathcal B}) \big \} \, . \end{aligned}$$

(131)

Here we shall use the semi-norms

$$\begin{aligned} |u|_{k, {\mathcal B}} := \sqrt{ \sum _{\alpha = k} | {\text {D}}^{\alpha } u |^2_{L^2 ({\mathcal B})} } , \qquad k = 0, 1, \ldots , s \, , \end{aligned}$$

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and the norm

$$\begin{aligned} \Vert u\Vert _{s, {\mathcal B}} := \sqrt{ \sum _{k \le s} |u|^2_{k, {\mathcal B}} } . \end{aligned}$$

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Critism: This expression for the norm does not take into account a typical length scale l of the problem, i.e., we are adding, for example, a square integrable function $|u|^2_{L^2 ({\mathcal B})}$ and its square integrable derivative $|u^{\prime }|^2_{L^2 ({\mathcal B})}$. Without any physically meaningful parameters these expression is hardly to interpret. This could be avoided by using the expression

$$\begin{aligned} \Vert u\Vert _{s, {\mathcal B}} := \sqrt{ \sum _{k \le s} l^{\text {d} k} \; |u|^2_{k, {\mathcal B}} } \, , \end{aligned}$$

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where d characterizes the dimension of ${\mathcal B}\subset \mathbb {R}^{\text {d} }$, Boffi et al. (2013).

With ${\text {D}}^{\alpha }$ as the $\alpha$-st weak differential operator. Thus the often used spaces $H^1 ({\mathcal B})$ and $H^1_0 ({\mathcal B})$ are defined by

$$\begin{aligned} H^1 ({\mathcal B}) = \big \{ u + {\text {D}}^1 u \in L^2 ({\mathcal B}) \big \} \, , \end{aligned}$$

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and

$$\begin{aligned} H^1_0 ({\mathcal B}) = \big \{ u \in H^1 ({\mathcal B}) \; : \; u = 0 \;\; \text {on}\;\; \partial {\mathcal B}\big \} \, . \end{aligned}$$

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For completeness we introduce the spaces $H^2 ({\mathcal B})$ and $H^2_0 ({\mathcal B})$ defined by

$$\begin{aligned} H^2 ({\mathcal B}) = \big \{ u + {\text {D}}^1 u + {\text {D}}^2 u \in L^2 ({\mathcal B}) \big \} \, , \end{aligned}$$

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and

$$\begin{aligned} H^2_0 ({\mathcal B}) = \big \{ u \in H^2 ({\mathcal B}) \; : \; u = 0 \;\;\text {and} \;\; \dfrac{\partial u}{\partial n} = 0 \;\;\text {on}\;\; \partial {\mathcal B}_u \big \} \, . \end{aligned}$$

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For negative superscripts, i.e., $H^{-s} ({\mathcal B})$ with $s > 0$, the spaces are identified with the duals of $H^{s}_0 ({\mathcal B})$:

$$\begin{aligned} H^{-s} ({\mathcal B}) = (H^{s}_0 ({\mathcal B}))^{\prime } \, . \end{aligned}$$

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For example, the norm associated to $H^{-1} ({\mathcal B})$, which is the dual of $H^{1}_0 ({\mathcal B})$, is defined as

$$\begin{aligned} \Vert u\Vert _{-1, {\mathcal B}} = \underset{v \in H^1_0 ({\mathcal B}) \setminus 0}{\text {min}} \dfrac{ (u, v)_{0, {\mathcal B}} }{\Vert v\Vert _{1 , {\mathcal B}}} \, . \end{aligned}$$

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The norm associated to $H^{-{}^{1}\!/_{2}} (\partial {\mathcal B})$, the dual of $H^{{}^{1}\!/_{2}}_0 (\partial {\mathcal B})$, is defined as

$$\begin{aligned} \Vert u\Vert _{-{}^{1}\!/_{2}, \partial {\mathcal B}, 0} = \underset{v \in H^{{}^{1}\!/_{2}} (\partial {\mathcal B}) \setminus 0}{\text {min}} \dfrac{ (u, v) }{\Vert v\Vert _{{}^{1}\!/_{2}, \partial {\mathcal B}}} \, . \end{aligned}$$

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The Hilbert space $H_0^{m} ({\mathcal B})$ is a closed subspace of $H^{m} ({\mathcal B})$; furthermore is $H_0^{0} ({\mathcal B}) = L_{2} ({\mathcal B})$.

$$\begin{aligned} \begin{array}{ccccccccccc} \ldots &{} H^{-2} ({\mathcal B}) &{} \supseteq &{} H^{-1} ({\mathcal B}) &{} \supseteq &{} L_{2} ({\mathcal B}) &{} \supseteq &{} H^{1}_0 ({\mathcal B}) &{} \supseteq &{} H^{2}_0 ({\mathcal B}) &{} \ldots \\ \ldots &{} \Vert u\Vert _{-2, {\mathcal B}} &{} \le &{} \Vert u\Vert _{-1, {\mathcal B}} &{} \le &{} \Vert u\Vert _{0, {\mathcal B}} &{} \le &{} \Vert u\Vert _{1, {\mathcal B}} &{} \le &{} \Vert u\Vert _{2, {\mathcal B}} &{} \ldots \end{array} \end{aligned}$$

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For tensorial Sobolev spaces, e.g., the three-dimensional tensor product space

$$\begin{aligned} H^{s} ({\mathcal B}) \times H^{s} ({\mathcal B}) \times H^{s} ({\mathcal B}) \end{aligned}$$

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we use the abbreviation

$$\begin{aligned}{}[H^{s} ({\mathcal B})]^3 = \prod _{i = 1}^{3} H^{s} ({\mathcal B}) \qquad \text {and analogously}\qquad [L^2 ({\mathcal B})]^3 = \prod _{i = 1}^{3} L^2 ({\mathcal B}) \, . \end{aligned}$$

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Let ${{\varvec{u}}}\in \mathbb {R}^3$ and set the Hilbert space

$$\begin{aligned} H (\text {div}; {\mathcal B}) = \big \{ {{\varvec{u}}}\in [L^2 ({\mathcal B})]^3 \; : \; {\text {div}}{{{\varvec{v}}}} \in L^2 ({\mathcal B}) \big \} \, , \end{aligned}$$

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with the associated norm

$$\begin{aligned} \Vert {{\varvec{v}}}\Vert _{H(\text {div}; {\mathcal B})} = \big \{ \Vert {{\varvec{v}}}\Vert ^2 + |{\text {div}}{{{\varvec{v}}}}|^2 \big \}^{{}^{1}\!/_{2}} \, . \end{aligned}$$

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