Discontinuous Galerkin methods for elliptic problems

Discontinuous Galerkin methods for elliptic problems#

Derivation of the Symmetric Interior Penalty method#

Any norm \(\|\cdot\|_{\mcP_h}\) used in this work which involves a collection of geometric entities \(\mcP_h\) should be understood as broken norm defined by \(\|\cdot\|_{\mcP_h}^2 = \sum_{P\in\mcP_h} \|\cdot\|_P^2\) whenever \(\|\cdot\|_P\) is well-defined, with a similar convention for scalar products \((\cdot,\cdot)_{\mcP_h}\). Any set operations involving \(\mcP_h\) are also understood as element-wise operations, e.g., \( \mcP_h \cap U = \{ P \cap U \st P \in \mcP_h \} \) and \( \partial \mcP_h = \{ \partial P \st P \in \mcP_h \} \) allowing for compact short-hand notation such as \( (v,w)_{\mcP_h \cap U} = \sum_{P\in\mcP_h} (v,w)_{P \cap U} \) and \( \|\cdot\|_{\mcP_h\cap U} = \sqrt{\sum_{P\in\mcP_h} \|\cdot\|_{P\cap U}^2}\).

The final symmetric interior penalty for the Poisson problem is:

Definition 14 (Symmetric Interior Penalty Method)

Find \(u_h \in V_h\) such that \(\foralls v_h \in V_h\)

(53)#\[ a_h(u_h,v_h) = l_h(v_h)\]

where the discrete bilinear form \(a_h(\cdot, \cdot)\) and linear form \(l_h(\cdot)\) are defined by

(54)#\[\begin{split} a_h(v,w) &= (\nabla v, \nabla w)_{\mcT_h} - (\partial_n v, w)_{\Gamma} - (v, \partial_n w)_{\Gamma} + \beta (h^{-1} v,w)_{\Gamma} \\ &\qquad - (\avg{\partial_n v }, \jump{w})_{\mcF_h} - (\jump{v}, \avg{\partial_n w })_{\mcF_h} + \beta (h^{-1}\jump{v},\jump{w})_{\mcF_h}, \\ l_h(v) &= (f,v)_{\mcT_h} - (\partial_n v, g)_{\Gamma} +\beta (h^{-1} g, v)_{\Gamma},\end{split}\]

respectively.

The theoretical analysis of the SIP method will be split into several parts.

Norms#

\[\begin{split}\| v \|^2_{a_h} &= \| \nabla v \|^2_{\mcT_h} + \|h^{-1/2} [v] \|^2_{\mcF_h} \\ \| v \|^2_{a_h, \ast} &= \| v \|_{a_h}^2 + \| h^{\onehalf} \avg{\partial_n v}\|_{\mcF_h}^2\end{split}\]

Discrete coercivity and boundedness#

  • Introduce norms

  • Discuss Lax-Milgram

  • Inverse trace inequalities

  • Discrete coercivity

Abstract error estimate#

Projection operators#

  • Abstract Cea’s Lemma

  • \(L^2\) projection

  • Scaled trace inequality and error estimates

A priori error analysis#

  • Energy error estimates

  • Adjoint consistency, Aubin-Nitsche trick and \(L^1\) error estimates

Advantages and disadvantages of dG methods#

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