Principal investigators
Prof. Dr. Willy Dörfler  (7/2015  )  
Prof. Dr. Christian Wieners  (7/2015  ) 
Project summary
For acoustic, elastic, and electromagnetic wave equations, we developed fully adaptive discretizations in space and time with integrated error control and efficient implicit solution methods with, at least numerically validated (almost) optimal complexity. The system can have still millions of degrees of freedom. Therefore it must be parallelized on several computational cores and solved using a spacetime multigrid preconditioner.
One motivation for developing spacetime methods is the design of modern computer facilities with an enormous number of processor cores, where the parallel realization of conventional methods becomes inefficient. Since these machines allow a fully implicit spacetime approach, new parallel solution techniques are required to solve the huge linear systems.
Linear wave equations

Acoustic waves: find pressure \(p\) and velocity \(\boldsymbol{v}\) with \[\partial_t p  \kappa \operatorname{div} \boldsymbol{v} = 0 \,, \qquad \rho\partial_t \boldsymbol{v}  \nabla p = \boldsymbol{f}\] for permeability \(\kappa\) and density \(\rho\).

Elastic waves: find stress \(\boldsymbol{\sigma}\) and velocity \(\boldsymbol{v}\) with \[\partial_t \boldsymbol{\sigma}  \boldsymbol{C} \boldsymbol{\varepsilon}(\boldsymbol{v}) = \boldsymbol{0} \,, \qquad \rho\partial_t \boldsymbol{v}  \boldsymbol{\operatorname{div}} \boldsymbol{\sigma} = \boldsymbol{f}\] with strain rate \(\boldsymbol{\varepsilon}(\boldsymbol{v}) = \operatorname{sym} (\mathrm{D} \boldsymbol{v})\) and the constitutive relation \[\boldsymbol{C}\boldsymbol{\varepsilon} = 2\mu\boldsymbol{\varepsilon} + \lambda\operatorname{trace}(\boldsymbol{\varepsilon})\textbf{1}\,.\]
We also realized spacetime methods for electromagnetic waves [Fin16]. and for viscoacoustic and viscoelastic waves.
Discretization
We consider two different discretizations.

Spacetime discontinuous Galerkin discretization (STDG)
The discretization uses the trial space \(V_h \subset \mathrm{H}^1(0,T;\mathrm{L}_2(\Omega;\mathbb{R}^m))\), which is discontinuous in space but continuous in time. The test space \(W_h \subset \mathrm{L}_2((0,T)\times\Omega;\mathbb{R}^m))\) is discontinuous in space and time. Continue reading. Collapse content.On a tensor product spacetime mesh with a fixed mesh in space and a time series \(0=t_0\lt t_1\lt\cdots\lt t_N = T\) with a local selection of polynomial degrees in space and time \(p_R\) and \(q_R\) in every cell, we set for the local test space \(W_{h,R} = \mathbb{P}_{q_R1}((t_{n1},t_n);\mathbb{R}^m)\otimes\mathbb{P}_{p_R}(K;\mathbb{R}^m)\). Then, the local ansatz spaces \(V_{h,R}= V_h_R\) take the form \[ V_{h,R} = \Big\{ \boldsymbol{v}_{h,R}\in \mathrm{L}_2(R;\mathbb{R}^m)\;\colon \boldsymbol{v}_{h,R}(t,\boldsymbol{x}) = \frac{t_nt}{t_nt_{n1}} \boldsymbol{v}_h(t_{n1},\boldsymbol{x}) + \frac{tt_{n1}}{t_nt_{n1}} \boldsymbol{w}_{h,R}(t,\boldsymbol{x})\,, \qquad\qquad\qquad\qquad\\ \quad\, \boldsymbol{v}_h \in V_h_{[0,t_{n1}]} \,,\ \boldsymbol{w}_{h,R} \in W_{h,R} \,,\ (t,\boldsymbol{x})\in R=I_n\times K \Big\} \,.\] The global test space is discontinuous in space and time and defined by \[W_h = \Big\{ \boldsymbol{w}_h\in \mathrm{L}_2((0,T);H)\;\colon \boldsymbol{w}_{h,R} = \boldsymbol{w}_h_R \in W_{h,R} \Big\} \,.\] The global ansatz space \(V_h\) is only discontinuous in space but continuous in time. More details are given in [DFWZ18]. Continue reading. Collapse content. 
Spacetime discontinuous PetrovGalerkin discretization (STDPG)
The Discontinuous PetrovGalerkin Method is a methodology to construct robust discretization schemes for linear PDEs. Due to its guaranteed stability independently of the mesh, it is well suited for adaptive applications. Moreover, as a LeastSquares type method, the resulting linear system is symmetric and positive definite which can be exploited by the iterative solver. Continue reading. Collapse content.In [Ern17], [EW18b], we construct a nonconforming variant of the DPG method for acoustic waves in spacetime and provide an framework to analyze it. Continue reading. Collapse content.
Numerical analysis
Using the variational setting, we can show infsup stability and a priori bounds for both discretizations.
The bilinear form \(b_h(\cdot,\cdot) = \big(L_h\cdot,\cdot)_{0,Q}\) is infsup stable in \(V_h\times W_h\) with \(\beta = C_1T^{1}\), i.e., \[\sup_{\boldsymbol{w}_h\in W_h\setminus\{\boldsymbol{0}\}} \frac{b_h(\boldsymbol{v}_h,\boldsymbol{w}_h)}{\\boldsymbol{w}_h\_W} \geq \beta\, \\boldsymbol{v}_h\_{V_h}\,,\qquad \boldsymbol{v}_h\in V_h\,.\] A direct result is that for given \(\boldsymbol{f}\in\mathrm{L}_2(Q;\mathbb{R}^J)\) a unique solution \(\boldsymbol{u}_h\in V_h\) exists solving \[(L_h\boldsymbol{u}_h,\boldsymbol{w}_h)_{0,Q} = (\boldsymbol{f},\boldsymbol{w}_h)_{0,Q}\,,\qquad \boldsymbol{w}_h\in W_h\] and satisfying the a priori bound \(\\boldsymbol{u}_h\_{V_h} \leq C_2T \M_h^{1}\Pi_h\,\boldsymbol{f}\_W\).

Numerical analysis for STDG Continue reading. Collapse content.
The discrete operator \(L_h\) is obtained by Galerkin approximation and upwind flux. For the proof, we refer to [Fin16], [DFW16] and [DFWZ18]. Continue reading. Collapse content. 
Numerical analysis for STDPG Continue reading. Collapse content.
An extensive numerical study in [Ern17] demonstrates that the theoretically predicted convergence rates are achieved in practice for various benchmark problems in one and two spatial dimensions.
While the analysis of DPG in [Ern17], [EW18b] is restricted to homogeneous materials in space, [EW18a] extends the arguments to the case of inhomogeneous material distributions. Continue reading. Collapse content.
Numerical experiments

Numerical experiment with STDG
The first example illustrates seismic tunnel exploration: An artificially generated surface wave (at \(x_\text{mid}\)) in the tunnel propagates into the solid and the reflected waves are measured in a certain region (marked red in the figure). We solve the acoustic wave equation in two space dimensions using a spacetime discontinuous Galerkin discretization. In space we consider discontinuous polynomials over quadrilaterals of degree \(p\) and continuous polynomials of degree \(q\) in time (for details see [DFWZ18]). On the right we see a sketch of the computational domain \(\Omega\). The region of interest is marked in red. Starting with a finite volume discretization \((p=0, q=1)\) we perform four adaptive refinement steps using a dual weighted goaloriented error estimator.The animation shows the spacetime solution being sliced in time. On the bottom is the polynomial degree with 0 (blue) to 4 (red). The adaptive algorithm is controlled by a goal functional with support in the region of interest.
We compared the uniform and adaptive refinement in the acoustic case observe that the adaptive algorithm saves 70% of the degrees of freedom while achieving the same accuracy compared with uniform refinement. Continue reading. Collapse content.refstep #DoF (effort) GMRES steps with MGPC \(\vartriangle\,E_{\text{ex}}\) in % uniform refinement r=1 534 528 7 1.85 r=2 2 405 376 13 0.22 r=3 6 414 336 19 0.49 r=4 13 363 200 27 0.25 adaptive refinement r=0 133 632 5 91.01 r=1 291 411 (55%) 7 1.27 r=2 819 279 (34%) 13 0.58 r=3 1 875 753 (29%) 20 0.56 r=4 3 594 969 (27%) 28 0.38 Acoustic wave: Uniform vs. adaptive refinement on \(44\,544 =928 \times 48\) spacetime cells distributed on 64 processor cores. The error \(\vartriangle E_\text{ex}(\mathbf{u_h}) = E(\mathbf{u_h}) E_\text{ex}\) of the goal functional is approximately estimated with respect to a linear extrapolation of the uniform results.
The parallel scaling behavior of the parallel multilevel preconditioner is tested for different numbers of processes. On mesh level 4 we have 2 850 816 spacetime cells, a linear discretization in space and time results in 34 209 792 DoFs for the acoustic case. The computing time for solving this huge linear system system with the parallel multigrid method scales nearly optimal. Show more...
For further explanations can be found in [DFWZ18] and a introduction into the used software in [Wie16] and Summer school SpaceTime.

Numerical experiment with STDPG
Here, we also demonstrate that the spacetime DPG method can be applied successfully to an application driven benchmark problem that has been inspired by the Marmousi benchmark. For this benchmark, we use the DPG method with tensorproduct polynomials of order \(k\) in each spacetime cell and tensorproduct elements of order \(k+1\) on the spacetime interfaces.
The Marmousi model is a two space dimensions seismic model devised by the Institut Français du Petrole to test seismic data inversion.
In contrast to the \(pq\)adaptive algorithm used with STDG, we decided to reduce the computational effort by truncating the spacetime mesh.
The next animation shows, that we don't lose any informations by truncating the mesh. We also verify the results by comparing them with STDG and a classical time stepping scheme.
The next animation shows, that we don't lose any informations by truncating the mesh. We also verify the results by comparing them with a classical time stepping scheme (TSDG).
Outlook
We also consider nonlocal material laws in time (e.g., acoustic and elastic waves with attenuation) and, as long term goal, this will be extended to the numerical homogenization of materials with (random) heterogeneous microstructure. We also consider heterogeneous media on a mesoscopic scale by constructing a local spacetime multiscale basis on finer meshes resulting into a hierarchy of multiscale models.