Trefftz discontinuous Galerkin discretization for the Stokes problem (Q6562910)

The authors consider an open, bounded and Lipschitz domain \(\Omega \subset \mathbb{R}^{d}\), with \(d=2,3\), and the Stokes equations: \(-\nu \Delta u+\nabla p=f\), \(-\operatorname{div}u=g\), in \(\Omega \), with the homogeneous Dirichlet boundary conditions \(u=0\) on \(\partial \Omega \), where \(f,g\) are external body forces and \(\nu >0\) is the dynamic viscosity.\N\NThe purpose of the paper is to propose a Trefftz-discontinuous Galerkin formulation for the Stokes problem. Let \(\mathcal{T}_{h}\) be a sequence of shape-regular simplicial triangulations of the polygonal domain \(\Omega \), \(\mathcal{F}_{h}\) the set of facets in the mesh \(\mathcal{T}_{h}\), \(\mathcal{P}^{k}(S)\) the space of polynomials up to degree \(k\) on an domain \(S\), \(\mathcal{P}^{k}=\mathcal{P} ^{k}(\mathbb{R}^{d})\), \(\mathbb{P}^{k}(\mathcal{T}_{h})\) and \(\mathbb{P}^{k}( \mathcal{F}_{h})\) the broken, i.e., element- or facet-wise polynomial space, such that for instance for \(v\in \mathbb{P}^{k}(\mathcal{T}_{h})\) it holds \( v\mid _{T}\in \mathcal{P}^{k}(T)\), for all \(T\in \mathcal{T}_{h}\). The Trefftz-discontinuous Galerkin formulation for the Stokes problem consists in finding \((u_{h},p_{h})\in \lbrack \mathbb{P}^{k}]^{d}\times \mathbb{P}^{k-1}/ \mathbb{R}\), such that: \(a_{h}(u_{h},v_{h})+b_{h}(v_{h},p_{h})=(f,v_{h})_{ \mathcal{T}_{h}}\), \(\forall v_{h}\in \lbrack \mathbb{P}^{k}]^{d}\), \( b_{h}(u_{h},q_{h})=(g,q_{h})_{\mathcal{T}_{h}}\), \(\forall q_{h}\in \mathbb{P} ^{k-1}/\mathbb{R}\), with the bilinear forms \N\[a_{h}(u_{h},v_{h})=(\nu \nabla u_{h},\nabla v_{h})_{\mathcal{T}_{h}}-(\{\nu \partial _{n}u_{h}\},[[v_{h}]])_{\mathcal{F}_{h}}-(\{\nu \partial _{n}v_{h}\},[[u_{h}]])_{\mathcal{F}_{h}}+\frac{\alpha \nu }{h} ([[u_{h}]],[[v_{h}]])_{\mathcal{F}_{h}},\] \N\[b_{h}(v_{h},p_{h})=-(\operatorname{div}v_{h},p_{h})_{\mathcal{T}_{h}}+([[v_{h}]\cdot n]],\{p_{h}\})_{\mathcal{F}_{h}},\] \Nwhere the interior penalty parameter \( \alpha =\mathcal{O}(k^{2})\) is chosen sufficiently large and \(\partial _{n}u=\nabla u\cdot n\). Here \([[v_{h}]]=v_{T}-v_{T^{\prime }}\) and \( \{v_{h}\}=\frac{1}{2}(v_{T}+v_{T^{\prime }})\), \(T\) and \(T^{\prime }\) being two neighboring elements sharing a common facet \(F\in \mathcal{F}_{h}^{i}\), the subset of interior facets, where \(T\) and \(T^{\prime }\) are uniquely ordered. \N\NIntroducing the space \(\mathbb{X}_{h}(\mathcal{T}_{h})=[\mathbb{P} ^{k}(\mathcal{T}_{h})]^{d}\times \mathbb{P}^{k-1}(\mathcal{T}_{h})/\mathbb{R} \), the authors gather the above formulations in a single one: \( K_{h}((u_{h},p_{h}),(v_{h},q_{h}))=(f,v_{h})_{\mathcal{T}_{h}}(g,q_{h})_{ \mathcal{T}_{h}}\), \(\forall (v_{h},q_{h})\in \mathbb{X}_{h}\). They recall properties of the spaces \(\mathbb{T}_{f,g}^{k}(\mathcal{T} _{h}):=\{(u_{h},p_{h})\in \mathbb{X}_{h}^{k}(\mathcal{T}_{h})\mid -\Delta u_{h}+\nabla p_{h}=\Pi ^{k-2}f\), \(-\operatorname{div}u_{h}=\Pi ^{k-1}g\) on \(\mathcal{T} _{h}\}\), where \(\Pi _{S}^{k}:L^{2}(S)\rightarrow \mathbb{P}^{k}(S)\) denotes the \(L^{2}\) projection into the scalar-valued polynomial space of order \(k\) on \(S\). The authors prove surjectivity properties of the Laplacian and Stokes operators between spaces \(\mathcal{P}^{k}\), from which they deduce the dimensions of the local finite element spaces \(\mathbb{X}_{h}(T)\) and \( \mathbb{T}(T)\). They propose an a priori error analysis. They prove a Ladyzhenskaya-Babuška-Brezzi stability result. Finally the authors build a best approximation result in the Trefftz space and prove an Aubin-Nitsche-like result for the \(L^{2}\)-error of the velocity. \N\NThe paper ends with the presentation of numerical simulations concerning two examples. In the first one, an exact solution exists, while the second one deals with a wedge that produces an infinite amount of Moffat eddies that differ greatly in magnitude. The authors test different methods involving different spaces. Considering simplicial meshes, they compare the total number of degrees of freedom, the number of degrees of freedom that remain after elimination of all interior unknowns, and the resulting sparsity pattern of the methods (after static condensation) in terms of the non-zero entries in the resulting sparse system matrices. They observe that the Trefftz-discontinuous Galerkin method brings a significant improvement over the standard discontinuous Galerkin method, especially in terms of non-zero entries in the matrix.