A PRIORI ERROR ESTIMATES FOR THE SPACE-TIME FINITE ELEMENT APPROXIMATION OF A QUASILINEAR GRADIENT ENHANCED DAMAGE MODEL

In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of a quasilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of two quasilinear elliptic PDEs which have to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. Numerical experiments are added to illustrate the proven rates of convergence. Mathematics Subject Classification. 65J08, 65M12, 65M15, 65M60. Received May 6, 2020. Accepted April 26, 2021.


Introduction
In this paper, we derive a priori error estimates for the space-time finite element discretization of a quasilinear gradient enhanced damage model. To be more specific, we investigate the finite element approximation of the quasilinear damage model The function : R → [ , 1] should belong to 2 (R) with ′ , ′′ ∈ 0,1 (R) ∩ ∞ (R) and > 0. Nemytskiioperators associated to will again be denoted by . The fourth-order tensor C ∈ ∞ (Ω; ℒ(R 2×2 sym )) is assumed to be symmetric and uniformly coercive, i.e., there exists a constant C > 0 such that C( ) : ≥ C | | 2 ∀ ∈ R 2×2 sym and f.a.a. ∈ Ω.
For isotropic materials, the material is completely described by the two Lamé parameters and which depend on Poisson's ratio and Young's modulus . Hooke's law states that the stress depends linearly on the strain = C = trace( ) + 2 .
We start with a time-independent variant of the PDE-system (1.1)-(1.2).

Fréchet derivatives of and
For the estimation of the spatial discretization error, we will need Lipschitz continuity of the Fréchet derivatives of and with respect to . Therefore, we have a look at partial derivatives. Lemma 2.8. There exists an index ∈ (2, ) such that for every ∈ −1, (Ω) the map ( , ·) : 1 (Ω) → 1, (Ω) is Fréchet differentiable and, for all , ∈ 1 (Ω), the partial derivative fulfills Proof. We will only prove equation (2.12) as the first part has already been established in Lemma 5.3 of [25] and [32], p. 101. Due to the definition of the partial derivative w.r.t. we have for arbitrary 1 , 2 , ∈ 1 (Ω) We will estimate both terms separately. We test the first two terms with ∈ 1, ′ (Ω) to obtain where we employed the Lipschitz continuity of ′ , ‖ ( , 1 )‖ 1, (Ω) ≤ according to (2.7) for fixed and Hölder's inequality with 1 = 2 + 1 and 1 = 1 + 1 which leads to 1 = 1 + 1 for the second to last estimate. For the last estimate, we made use of the Lipschitz continuity of as fulfills the condition, see (2.8). Thus, we obtain (2.13) For the second term, we start with an upper bound for ( , 2 ) . Hölder's inequality yields since ≥ 2 −2 implies ∈ [2, ] and (2.7) is valid. Next, we have for 1 = 1
For a Lipschitz continuous derivative, it is easy to show that we have Note further, that ( , ) is symmetric, that is we have

Regularity of solutions
For the error estimation, the regularity given by Theorem 2.7 is not satisfying. On the one hand, we need more temporal regularity. We have to assume more temporal regularity for for the temporal error estimation, that is, we assume ∈ 1 ( ; 2 (Ω)) ˓→˓→ ∞ ( ; −1, (Ω)). This is due to the fact that the solution of the PDE system is only as regular in time as the right-hand side. On the other hand, we also require more spatial regularity for semidiscrete solutions. As these are solutions of time-independent elliptic equations we have a look at the regularity of solutions and given by the Definitions 2.4 and 2.6 next.
Higher regularity for solutions to elliptic systems in divergence form with mixed boundary conditions has been studied in a recent paper [18]. Without going into detail here (for details, we refer to [18], Sect. 5), we require that ( )C is a multiplier on (Ω, R 2 ) for some > 0. A sufficient condition for this property is Hölder continuity according to Lemma 1 of [18].
Thus, we have a look at the regularity of = Φ( , ) first. We assume ∈ 1, (Ω) and ∈ 1 (Ω) since according to Theorem 3.12 of [19], the semidiscrete solution is more regular in space. It is as regular as (which at the moment and according to Theorem 2.7 belongs to 1 (Ω)) as long as the regularity is less than 3 2 (Ω). Proposition 2.10. Let ∈ 2 (Ω) and ∈ 1 (Ω) be given. Then, the solution = Φ( , ) given by Definition 2.6 belongs to 2, 2 (Ω) ˓→ 0, (Ω) and thus is Hölder continuous with Hölder exponent Proof. The proposition follows with standard argumentation from elliptic regularity theory.
Thus, we have Hölder continuity of ( ) since is Hölder continuous and is Lipschitz continuous which yields that ( ) is indeed a multiplier. For C, we could also impose Hölder continuity. But since C depends on material parameters, for which continuity might be to restrictive, we just assume the following Assumption 2.11. From now on, we assume that C is a multiplier on (Ω, R 2 ) for some > 0.
Note, that Remark 5 of [18], ensures that Assumption 2.11 is fulfilled for example for piecewise constant, discontinuous material parameter functions. Based on this assumption we arrive at the following regularity result Lemma 2.12. With Assumption 2.11, for a given right-hand side ∈ 2 (Ω) and a function ∈ 0, (Ω), ∈ (0, 1) the solution of the equation −div( ( )C ( )) = admits the additional regularity ∈ 1+ (Ω) for a fixed ∈ (0, 1) and fulfills with a constant > 0 depending on the multiplier norm of .
Remark 2.13. Lemma 2.12 only ensures the existence of such that ∈ 1+ (Ω) but the relation of and the integrability > 2 obtained by 1, -theory is still an open question. For this contribution, we assume that the integrability obtained by 1, -theory cannot be improved any further. Since ∈ 1+ (Ω) ˓→ holds true for the largest possible obtained by 1, -theory. In case there is no largest possible , the above inequality will be assumed for the supremum˜∈ (2, ∞]. Based on Proposition 2.10 we state the improved regularity of Lemma 2.14. Let Assumption 2.11 be fulfilled. Moreover, let ∈ 2 (Ω) and ∈ 1 (Ω) be given. Then, the solution given by Definition 2.6 belongs to 2, 2 (Ω) ˓→ 1+ (Ω), = min{1, 2 − 4 } and fulfills with a constant > 0 independent of and .
Finally, to be able to apply duality arguments, we need to know the regularity of solutions to linearized equations. Since = is already linear in , we only have a look at solutions of Lemma 2.15. Let Assumptions 2.1 and 2.11 be fulfilled. Moreover, let ∈ 2 (Ω), ∈ 2 (Ω), = Φ( , ) and ∈ (Ω) be given. Then, the variational problem with a constant > 0 only depending on ‖ ‖ 2 (Ω) and with = min{ , 2 }.
Proof. This is a standard result from elliptic theory.

Semidiscretization in time
Now, we turn our attention towards the a priori error estimates for a discrete version of the damage model. We split the errors in a temporal and a spatial part and start with the error in time.

Discretization in time
For the discretization in time we will employ discontinuous constant finite elements. Therefore, we consider a partition of the time interval = [0, ] as ] of length and time points We set := max{ : = 1, ..., }. The semidiscrete trial and test spaces are given as We use the notation To express the jumps possibly occurring at the nodes we define Note, that for functions piecewise constant in time the definition reduces to Then, the semidiscrete problem is given as follows: Find states ( , , ) ∈ 0 ( ) × 0 × 0 such that We require the interpolation/projection onto 0 , 0 and 0 , respectively. Therefore, we define the semidiscrete interpolation operator : ( ; 2 (Ω)) → 0 with | ∈ P 0 ( ; 2 (Ω)) via ( )( ) = ( ) for = 1, ..., . For the projection we employ the standard 2 -projection in time : 2 ( ; 2 (Ω)) → 0 given by | := 1 ∫︀ ( )d . Both operators will always be denoted by the same symbols despite possibly different domains and ranges. Note, that integration in time preserves spatial regularity due to the definition of the Bochner integral, that is has the same spatial regularity as the preimage . Note furthermore, that the projection is ( ; )-stable for all ∈ [1, ∞] and any arbitrary Banach space .

Temporal error estimates
We will prove linear convergence in time in suitable norms for the displacement as well as for the two damage variables. In accordance with our discretization technique, we may choose test functions vanishing outside the subinterval . Then, the solution of (3.1)-(3.3) also fulfills for ∈ 0 ( ′ ), ∈ 0 and for all = 1, . . . , . This system is equivalent to (Ω), ∈ 1 (Ω) and for all = 1, . . . , . Note, that | ∈ −1, (Ω) according to the preserved spatial regularity of a temporal projection. We easily see that the solution of the now time-independent elliptic PDE system is given as Since the reduced fixed point equation is identical to the fixed point equation in [19] (up to a different solution operator Φ), existence of a unique solution , ∈ 2 (Ω) can be proven as in Proposition 3.1 of [19], by means of Banach's fixed point theorem.
Proof. The lemma can be proven along the lines of the proof of Lemma 3.2 of [19].
Proof. The lemma can be proven along the lines of the proof of Lemma 3.14 of [19], since Lemma 3.2 and Theorem 3.1 imply that Remark 3.4. Note, that Lemma 3.2 ensures that the constant 1 from Theorem 3.1 is only dependent of ‖ ‖ 1 ( ; 2 (Ω)) .
We turn our attention to the derivation of temporal discretization error estimates. Due to the special definition of the semidiscrete solutions and , preliminary estimates for temporal errors are easily obtained from Lipschitz continuity of the solution operators and Φ. Indeed, we have and (3.14) Thus, a projection error estimate for and an error estimate for the temporal error of − directly lead to error estimates for − and − . We still have to decide which norm we want to choose for the temporal error estimation. The possible norms for and depend on the projection error estimates available for . Since we assume ∈ 1 ( ; 2 (Ω)) we choose 2 ( ; ·)-norms for and . But for , it is possible to derive ∞ ( ; 2 (Ω))-error estimates. In Ch. 2 of [32], Susu proves linear convergence of the implicit Euler method applied to the reduced ODE in ∞ ( ; 2 (Ω)). It is well known that the implicit Euler method follows from the piecewise constant discontinuous Galerkin method by approximating the right-hand side | by ( ). Therefore, both methods are highly linked and we want to establish error estimates in ∞ ( ; 2 (Ω)) as well. Note, that our proof differs from the method employed in [32] as we work with a semidiscrete solution which is discontinuous in time whereas Susu assumes continuity in time for the semidiscrete solution. Convergence of the implicit Euler method in ∞ ( ; 2 (Ω)) is a byproduct from the convergence in 1,∞ ( ; 2 (Ω)) which is proven in [32] whereas we will prove convergence directly. Note further, that Susu also provides an error estimate for − in the ∞ ( ; 2 (Ω))-norm. This is possible as she assumes Lipschitz continuity with respect to time for the control , i.e. such a result requires ∈ 1,∞ ( ; −1, (Ω)).
Proof. The lemma can by proven by employing Gronwall's inequality. We rewrite and˜as for almost all ∈ [0, ]. Due to the Lipschitz continuity of max and Φ and Lem. 2.2, we obtain

As ( ) is constant and ( ) is monotone increasing, Gronwall's inequality yields the desired result.
The second part of the error will also be proven with Gronwall's inequality. But, as˜− is continuous in time only on each subinterval , we have to split the overall time horizon into the subintervals.
Proof. At first, we have according to Theorem 3.1 and Lemma 3.2 with a constant˜> 0 depending (among others) on and 0 , but not on . We will need this estimate later. Next for ∈ , = 1, . . . , , we rewrite (3.17) as and (3.16) as Gronwall's lemma then yields The assertion now follows as the right-hand side is independent of ∈ .
Finally, we are able to state an a priori error estimate for − .
Theorem 3.7. For the error between the continuous solution ∈ 1,∞ ( ; 2 (Ω)) and the semidiscrete solution ∈ 0 for a given load ∈ 1 ( ; 2 (Ω)) and a given initial state 0 ∈ 2 (Ω), we have the error estimate Proof. We just need to combine the results of Lemmas 3.5 and 3.6. This gives us The error estimate for˜− in the ∞ ( ; 2 (Ω))-norm is also an error estimate for the 1 ( ; 2 (Ω)) norm and therefore this term also converges linear. Due to ∈ 1 ( ; 2 (Ω)) we have Thus, we arrive at the desired result.

Discretization in space
In this section, we provide error estimates for the spatial discretization.

Spatial error estimates
In contrast to temporal error estimates, spatial error estimates are not derived that easily. We have to derive spatial error estimates in several steps. At first, we establish preliminary error estimates for − ℎ and − ℎ in 1 (Ω) and 2 (Ω) on each subinterval . In the next step, we show an error estimate for − ℎ in 2 ( ; 2 (Ω)) which leads to error estimates for − ℎ and − ℎ in 2 ( ; 1 (Ω)) and 2 ( ; 2 (Ω)) as well.

1, ′
(Ω), we have the property of Galerkin orthogonality, that is for all ∈ 1 ℎ . Thus, we obtain with the coercivity and boundedness of , Then, the assertion follows directly from the error estimate for the projection ℎ . This error estimate for ∈ 1+ (Ω) on 1,2 (Ω) can be obtained as in the case ∈ 1+ (Ω) by making use of appropriate interpolation spaces and a corresponding error for the usual interpolation operator, see for example section 4 and 14 of [2].
Thus, we have the overall error estimate with a constant > 0 independent of , ℎ and the subinterval .
Error estimates for − ℎ Next, we derive an error estimate for , − ℎ, in 1 (Ω) on each subinterval . The approach is similar to the 1 (Ω)-error for , − ℎ, . We may express (3.2) and (4.2) with and (see Def. 2.5) as While unique solvability of (4.9) is clear, we still have to take a look at the unique solvability of (4.2). It suffices to prove the existence of a unique solution ℎ, on each subinterval for a given right-hand side ℎ, ∈ 2 (Ω).
Lemma 4.4. Let Assumption 2.1 be fulfilled. For given ∈ −1, (Ω) and ∈ 2 (Ω) the equation possesses a unique solution ℎ ∈ 1 ℎ . Proof. Existence of a solution may be proven with Brouwer's fixed point theorem. Uniqueness then follows due to the strong monotonicity of + under Assumption 2.1.
2 ( ; 2 (Ω))-error estimates for − ℎ Error estimates for , − ℎ, and , − ℎ, thus only depend on an error estimate for , − ℎ, in 2 (Ω). We are going to establish such an error estimate in the following. Note, that the approach is highly linked to [19] as the problem discussed in the next paragraph and the problem discussed in Section 3.2 of [19], only differ in the PDE underlying the solution operator Φ. We thus only state important results.
The required regularity for as well as uniform boundedness in 2 ( ; (Ω)) is proven in Theorem 3.1 and Lemma 3.3. An estimate for the second term is given in [19] as well, see Lemma 3.9. The required preliminary error estimate for , − ℎ, in 2 (Ω) is given by Lemma 4.8.

Numerical example
In this last section, we have a look at a numerical example for the simulation of our damage model. We apply a fixed point argument to the reduced ODE to solve the model equations. In each time step, the solutions of the PDE system and are computed with a Newton solver. We work with isotropic materials and assume that our object is completely sound at the beginning of our observation, that is 0 = 0.
We consider an example with a non-fictitious material. It is based on a sharp notch problem presented in [21]. The geometry of the domain is illustrated in Figure 1  If the active set was empty at the beginning of the observation, damage would not occur since the load is chosen independent of time. Note, that our findings cannot be compared to the results from [21] since the authors chose a time-dependent displacement and different material parameters.
The exact solution is unknown. For each experiment, we compute a reference solution (ˆ,ˆ,ˆ) on a fine temporal or spatial grid. In both cases, the temporal discretization parameter has to be chosen small since, otherwise, the fixed point argument fails. The reason for this is the choice of . We choose larger than 1 because numerical tests show that the Newton solver cannot compute a solution of the PDE system otherwise. Therefore, this example also illustrates the theoretical results of [32] about the unique solvability of the PDE system. Figure 2 depicts the displacement and the development of both damage functions. The object is rotated by 180 ∘ for the visualization of the damage. The pictures of the nonlocal damage variable in the middle row show that there are three areas where damage might occur: the singular corner in the middle of the bottom boundary, the bottom right corner and the area in the middle of the top boundary. But exceeds the threshold only at the singular corner. Consequently, damage occurs in this area only . The bottom row depicts the development of the local damage .
For the visualization of the behavior of the temporal error, we compute the reference solution (ˆ,ˆ,ˆ) for ℎ = 3605 and = 2 12 . For the visualization of the behavior of the spatial error, the reference solution is computed for ℎ = 56684 and = 2 10 . We denote errors by ℎ :=ˆ− ℎ . Table 1 depicts the development of the temporal discretization errors. We observe first order convergence for all three functions. Tables 2 and 3 depict the development of the spatial 2 ( ; 2 (Ω))-and 2 ( ; 1 (Ω))-errors. At first we note, that the results are not perfect. There are some values for the experimental order of convergence (marked in italic) that are outliers. They were omitted from the calculation of the mean. Nonetheless, the results underline our theoretical findings of Theorem 4.13. We observe a rate of convergence in 2 ( ; 1 (Ω)) that is less than 1 for the errors in and . This rate is approximately doubled when we consider 2 ( ; 2 (Ω))-errors. All 2 ( ; 2 (Ω))-errors converge with a rate that is less than 3 2 . And finally, the 2 ( ; 2 (Ω))-errors for and converge with the same rate ( Fig. 3-5).   Table 2. 2nd example: 2 ( ; 2 (Ω))-errors for the states for a spatial refinement.  Table 3. 2nd example: 2 ( ; 1 (Ω))-errors for the states for a spatial refinement.