{'fraction_non_alphanumeric': 0.036023742921470966, 'fraction_numerical': 0.009892883946237293, 'mean_word_length': 4.650732459521974, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Space-time error control using a partition-of-unity dual-weighted residual method applied to low mach number combustion

In this work, a space-time scheme for goal-oriented a posteriori error estimation is proposed. The error estimator is evaluated using a partition-of-unity dual-weighted residual method. As application, a low mach number combustion equation is considered. In some numerical tests, different interpolation variants are investigated, while observing convergence orders and effectivity indices between true errors (obtained on a sufficiently refined mesh) and the error estimator.


Introduction
This work is devoted to space-time goal-oriented a posteriori error control. Such space-time schemes for error estimation and adaptivity in time, space, or both, are of current interest with various applications in parabolic problems [14], incompressible Navier-Stokes equations [3,4], dynamic Signorini, obstacle, and hyperbolic problems [6,5,12] and fluid-structure interaction [9,10].
Our method is based on prior work, in which the dual-weighted error estimator is realized within a weak formulation using a partition-of-unity [13]. We note that another weak realization is achieved with the so-called filtering approach [7], which was already applied in [14,3,4] to space-time error control and adaptivity.
However, the extension of the partition-of-unity (PU) localization to space-time settings has not yet been established in the published literature. We provide a detailed algorithmic derivation of the error estimator and discuss the important ingredients. As numerical example, we consider a nonlinear low mach number combustion problem. A key part is the backward-in-time running adjoint problem.
One difficulty is that, due to Galerkin orthogonality, the adjoint problem must contain higher order information (see [2]) in the primal error part and vice vera in the adjoint error part. Recently, for stationary settings a new class of algorithms could be established by using interpolation information in a smart way [8]. For our proposed space-time setting we investigate the performance by simply choosing different combinations for the spatial finite elements, such as low-order cG(1)/cG(1) and high-order equal elements cG(2)/cG (2) as well as the natural approach cG(1)/cG(2) (low order primal and higher order adjoint for the primal error part). These choices are investigated with respect to their convergence properties and evaluation of the effectivity indices. We will, however, limit ourselves to the variation of finite element orders in space and use the equal order approach for the temporal discretization. We also notice that some preliminary results on space-time adaptivity with the PU-DWR method are published in [15].
The outline of this work is as follows: In Section 2, the low mach number combustion equations are introduced and their weak formulation is provided. Next, in Section 3, the discretization with finite elements is described. In the main Section 4 a space-time PU-DWR error estimator is derived in detail. Finally, in Section 5 some numerical experiments are reported, that show the performance of our developments. The code for these simulations is based on our extension of the package dwr-diffusion [11] to solve nonlinear problems. The package itself uses deal.II [1] as the finite element library. We conclude our work in Section 6.

The Low Mach Number Combustion Equations
The nonlinear parabolic problem we want to investigate describes a combustion reaction under the low Mach number hypothesis. Under that hypothesis the dimensionless temperature θ and the concentration of the combustible species Y are not influencing the fluid velocity field.
For the special but important case of v = 0 all convection terms vanish and θ and Y are only influenced by diffusion and by the reaction mechanism in which Y combusts and raises the temperature.
For constant diffusion coefficients we arrive at the following set of equations where the reaction is described by Arrhenius law The parameters are the Lewis number Le = 1, the gas expansion α = 0.8 and the nondimensional activation energy β = 10.
Part of the boundary Γ R ⊂ Γ := ∂Ω will be cooled. This can be described by a Robin condition ∂ n θ = −kθ. Following the standard procedure, we obtain the following space-time variational formulation of our problem. Find u = (θ, Y ) such that where (f, g) describes the space-time scalar product defined as As the homogeneous Neumann condition on Γ N is a natural condition it does not appear in the variational formulation. The inhomogeneous Dirichlet conditions on Γ D are imposed as usual and inserted into the finite element spaces.

Discretization
As we want to use different finite elements for the temporal and the spatial test-and trial functions we will start by partitioning [0, T ] into M subintervals I n = (t n−1 , t n ], with the discretization parameter k = t n − t n−1 . In time we will use piecewise constant dicontinuous elements φ k (t) ∈ dG(0). To be able to use different refined meshes over time, so called dynamic meshes, we will discretize Ω on each subinterval by a triangulation T n h . Using quadrilaterals (in two-dimensional configurations) for the spatial triangulation, we can use continuous finite element functions φ h (x) ∈ cG(p) of order p as test functions and trial functions. The fully discrete equations on each subinterval are then obtained by using φkh ∈ dG(0)cG(p) as test functions and trial functions in (4). For a more detailed look at the discretization and the corresponding finite element spaces see [14].

Space-Time PU-DWR Error Estimation
Denoting our quantity of interest by the goal functional J(u), we obtain the Lagrange functional for minimizing the error in said quantity as As a first order optimality condition we obtain the KKT (Karush-Kuhn-Tucker) system and with it an auxiliary adjoint problem. In summary, we then have Note that for nonlinear problems A(u, z) is a semilinearform that is linear in z and that F (z) is always a linear form. Thus, the directional derivative in direction φ w. r. t. to z, i. e. A z (z)(u, φ) coincides with A(u, φ). The same holds for linear goal functionals and linear problems resulting in the dual problem A(ψ, z) = J(ψ). Also note that the adjoint problem obtained by this derivation applies the temporal derivative to the test function. To rectify this, a partial integration in time is applied to . This results in a problem that runs backwards in time and has a final condition instead of an initial condition.

Error Estimation
Following Proposition (2.1) in [2] we obtain the error representation where R is a higher order remainder term. In many cases it is sufficient to approximate the error by only computing the primal residual i.e.
which is also called primal error estimator. Subsequently the second term in the error representation is called adjoint error estimator ρ * kh (u kh )(u − u kh , z kh ). Introducing the semidiscrete solutions u k and z k which are still continuous in space the primal error estimator can be further split into a temporal estimator ρ k and a spatial estimator ρ h

Practical Evaluation
As the exact solutions u and z are unknown, we have to further approximate them to calculate the error estimators. For the temporal primal estimator, we will construct a piecewise linear solution i (1) k z on each grid point by linear interpolation between the piecewise constant solutions z n and z n−1 in the interval I n .
For the spatial estimator we will look at three different approaches. The simplest approach is calculating u kh with dG(0)cG(1) elements and z kh with dG(0)cG(2) elements. Then, we assume z kh to be the approximation of the exact solution and interpolate it down into cG(1) in space obtaining i (2,1) h z kh . This interpolation should be included in most finite element packages.
For also calculating the dual estimator we also need an approximation for u, which can be obtained by approximating u kh and z kh with dG(0)cG(2) elements. Using the same interpolation as before we can approximate the discrete solutions asũ kh := i As this approach can be quite memory intensive, both u kh and z kh can be solved using dG(0)cG (1) elements. The approximation for z can then be obtained by combining neighbouring cG(1) elements into one large cG(2) patch with the operator i (2) 2h . The operator and the requirements for the mesh are described in [7]. Using those interpolations we obtain the following primal estimators for the different approaches for u kh ∈ dG(0)cG(1) and z kh ∈ dG(0)cG (1), for u kh ∈ dG(0)cG(1) and z kh ∈ dG(0)cG (2), for u kh ∈ dG(0)cG(2) and z kh ∈ dG(0)cG (2).
The corresponding dual estimators are obtained by the same interpolation operators, but applied to the primal solution u kh and inserted into the adjoint problem.

Variational PU Localization
For use in adaptive refinement we need to obtain indicators η n K or η n i for each cell K or DoF i on the time interval I n , such that We propose a DoF-wise partition of unity (PU) χ n i , with M n=1 i∈T h the simplest choice is χ n i ∈ dG(0)cG (1). Effectively, this leads to a spatial PU per time step, that is identical to the approach of [13] for stationary problems. The estimators are obtained by multiplying the directions in the derivatives of the Lagrangian with the PU, which leads to the localization of the original error representation (8): Finally, inserting the PU into the estimators described in the previous subsection yields the error indicators for each space-time DoF. The initial conditions are described by

Numerical Example
The functional of interest is the space-time averaged reaction rate with final time T = 60.
The initial grid is solved with k = 0.234375, resulting in M = 256 time intervals.

Comparison of Error Estimators
To compare the estimators without influence of the adaptivity, the following simulations were done with global refinement in space and time. Tables 1 -3     Comparing the results over all tables, we see that the equal order approaches perform relatively similar and better than the mixed order approach. Especially on lower refinement levels the cG(1)/cG (2) results are orders of magnitude above the actual error. On closer inspection the adjoint solutions get larger with each time step, which leads to the estimator being dominated by the indicators on the first few time intervals. As the codes for solving the cG(2) adjoint problems for cG(2)/cG(2) and cG(1)/cG (2) basically only differ in the u kh that is inserted in the assembly of the right hand side and the nonlinear part of the matrix, we surmise that the errors from inserting a lower order solution get amplified with each time step. This would also explain why this approach does not fail for stationary problems even on coarse meshes. In previous simulations we also saw that pairing the cG(1)/cG (2) approach with solving the adjoint with cG(1) elements in time led to worse results than the temporal equal order approach, even for the linear heat equation with the L 2 error as functional of interest. For adaptivity on dynamic meshes this is of course a considerable problem, as it is advisable to start with a coarse mesh to only capture the local behaviour of the solution/functional at each time interval.
When comparing the tables with each other, we see that for this problem the primal error estimator on itself performs better than the adjoint error estimator and is comparable to the full estimator.
Overall, the cG(1)/cG(1) approach is preferable as it is considerably cheaper to calculate compared to the cG(2)/cG(2) approach for which multiple linear systems need to be solved with ≈ 8N unknowns (one solve for the adjoint and several solves for the primal Newton solver steps) instead of ≈ 2N unknowns. Additionally, the primal solution vectors have to be kept either in RAM or on hard disk as the adjoint is solved backwards in time, so the cG(2)/cG(2) leads to a considerable increase in memory or storage demand. From a computational standpoint one can see why the cG(1)/cG (2) approach would be a nice tradeoff between accuracy and memory demand as only a single linear solve per time step has to performed on the larger set of unknowns.

Adaptive Results
When comparing the different estimators in the cG(1)/cG(1) approach the primal estimator is closest to the actual error, so we decided to use this estimator as a basis for an adaptive simulation. As we have DoF-based indicators we compute cell-wise indicators to use build-in refinement strategies in deal.II. These are obtained by simply adding the four spatial indicators of the cell vertices. As a refinement strategy we chose fixed fraction marking in which the indicators are sorted and the x% of cells with the largest indicators are marked for refinement. The same strategy is applied to the time intervals for which the indicators are calculated as the sum over all temporal DoF-indicators on the corresponding spatial triangulation. As fractions we chose 50% for the temporal and 33% for the spatial refinement which leads to roughly 1.5M time intervals and 2N spatial cells per time interval compared to 2M and 4N for global refinement. Figure 2 shows that the exact error converges faster for adaptive refinement, when comparing the number of primal DoFs.
To see if our novel localization approach works well in capturing the local behaviour of the goal functional, Figures 3 and 4 show the evolution of the reaction rate ω over [0, T ] and the corresponding meshes. In all timesteps the combustion reaction is captured well by the fine cells.Additionally, for time steps after the flame passed the cooled rods, there is also some refinement around the sharp corners, which is to be expected.

Conclusions
In this work, we developed a space-time goal-oriented a posteriori error estimator using a newly developed partition-of-unity dual-weighted residual localization. As model problem we considered a nonlinear low Mach number combustion problem. Specific emphasis was on different space-time finite element combinations for the primal and adjoint subproblems. Therein, we detected a better performance for equal-order combinations of cG(1)/cG(1) and cG(2)/cG(2) type in comparison to a cG(1)/cG(2) finite element. The reason has not yet been fully understood by us and needs further future investigations whether algorithmic or mathematical problems are the reason. Finally, some illustrations of locally adaptive meshes show that the error indicators obtained by our proposed method yield excellent findings in terms of resolving the local flame front.