Let be a separable Hilbert space, and let be a reflexive Banach space such that constitutes a Gelfand triple. The main motivation for this work is to identify the structural properties for the drift operator of the nonlinear SPDE
which allow to construct a space-time discretization of (1.1) for which optimal strong rates of convergence may be shown. Relevant works in this direction are [5, 6], where both, σ and are required to be Lipschitz. The Lipschitz assumption for the drift operator does not hold for many nonlinear SPDEs including the stochastic Navier–Stokes equation, or the stochastic version of general phase field models (including (1.2)) below for example. A usual strategy for a related numerical analysis is then to truncate nonlinearities (see e.g. ), or to quantify the mean square error on large subsets . As an example, the following estimate for a (time-implicit, finite element based) space-time discretization of the 2D stochastic Navier–Stokes equation with solution was obtained in ,
for all , where (respectively ) is such that for (respectively for ). We also mention the work  which studies a spatial discretization of the stochastic Cahn–Hilliard equation.
Let , , be a bounded Lipschitz domain. We consider the stochastic Allen–Cahn equation with multiplicative noise, where the process solves
where is an -valued Wiener process which is defined on the given filtered probability space ; however, it is easily possible to generalize the analysis below to a trace class Q-Wiener process. Obviously, the drift operator is only locally Lipschitz, but is the negative Gâteaux differential of and satisfies the weak monotonicity property
for some ; see Section 2 for the notation. Our goal is a (variational) error analysis for the structure preserving finite element based space-time discretization (2.5) which accounts for this structural property, avoiding arguments that exploit only the locally Lipschitz property of to arrive at optimal strong error estimate.
The existing literature (see e.g. ) for estimating the numerical strong error on problem (1.2) mainly uses the involved linear semigroup theory; the authors have considered the additive colored noise case, in which they have benefited from it by using the stochastic convolution, and then used a truncation of the nonlinear drift operator to prove a rate of convergence for a (spatial) semi-discretization on sets of probability close to 1 without exploiting the weak monotonicity of . In contrast, property (1.3) and variational arguments were used in the recent work , where strong error estimates for both, semi-discrete (in time) and fully-discrete schemes for (1.2) were obtained, which are of sub-optimal order for the fully discrete scheme in the case . In , a standard implicit discretization of (1.2) was considered for which it is not clear to obtain uniform bounds for arbitrary higher moments of the solution of the fully discrete scheme, thus leading to sub-optimal convergence rates above. In this work, we consider the modified scheme (2.5) for (1.2), and derive optimal strong numerical error estimate.
The subsequent analysis for scheme (2.5) is split into two steps to independently address errors due to the temporal and spatial discretization. First we exploit the variational solution concept for (1.2) and the semi-linear structure of to derive uniform bounds for the arbitrarily higher moments of the solution of (1.2) in strong norms; these bounds may then be used to bound increments of the solution of (1.2) in Lemma 3.2. The second ingradient to achieve optimal error bounds is a temporal discretization which inherits the structural properties of (1.2); scheme (4.1) is constructed to allow for bounds of arbitrary moments of in Lemma 4.1, which then settles the error bounds in Theorem 4.2 by using property (1.3) to effectively handle the nonlinear terms. We recover the asymptotic rate which is known for SPDEs of the form (1.1) when is linear elliptic. It is interesting to compare the present error analysis for the SPDE (1.2) with the one in  for a general SODE with polynomial drift (see [7, Assumptions 3.1, 4.1, 4.2]) which also exploits the weak monotonicity of the drift.
The temporal semi-discretization was studied as a first step rather than spatial discretization to inherit bounds in strong norms which are needed for a complete error analysis of the problem. The second part of the error analysis is then on the structure preserving finite element based fully discrete scheme (2.5), for which we first verify the uniform bounds of arbitrary moments of (cf. Lemma 5.1). It is worth mentioning that, if is a solution to a standard space-time discretization which involves the nonlinearity , then only basic uniform bounds may be obtained (see [10, Lemma 2.5]), as opposed to those in Lemma 5.1. Next to it, we use again (1.3) for the drift, in combination with well-known approximation results for a finite element discretization to show that the error part due to spatial discretization is of order where is the time discretization parameter and is the space discretization parameter (see Theorem 5.2). In this context, we mention the numerical analysis in  for an extended model of (1.2), where the uniform bounds for the exponential moments next to arbitrary moments in stronger norms are obtained for the solution of a semi-discretization in space in the case (see [11, Propositions 4.2, 4.3]); those bounds, together with a monotonicity argument are then used to properly address the nonlinear effects in the error analysis and arrive at the (lower) strong rate for the p-th mean convergence of the numerical solution.
2 Technical Framework and Main Result
Throughout this paper, we use the letter to denote various generic constants. Let , , with be a cube in . Let us denote and for . Problem (1.2) is then supplemented by the space-periodic boundary condition
Let respectively denote the Lebesgue respectively Sobolev space of R-periodic functions . Recall that functions in may be characterized by their Fourier series expansion, i.e.,
Below, we set for . Throughout this article, we make the following assumption on .
We have that σ, , and are bounded. Moreover, σ is Lipschitz continuous, i.e., there exists a constant such that
Definition 2.1 (Strong Variational Solution).
Fix , and . A -valued -adapted stochastic process is called a strong variational solution of (1.2) if satisfies -a.s. for all that
The following estimate for the strong solution is well known (),
2.1 Fully Discrete Scheme
Let us introduce some notation needed to define the structure preserving finite element based fully discrete scheme. Let be a uniform partition of of size . Let be a quasi-uniform triangulation of the domain . We consider the -conforming finite element space (cf. ) such that
where is the space of -valued functions on K which are polynomials of degree less or equal to 1. We may then consider the space-time discretization of (1.2): Let , where denotes the -orthogonal projection, i.e., for all
Find the -adapted -valued process such that -almost surely
Solvability for is again immediate via the Brouwer fixed point theorem.
We are now in a position to state the main result of this article.
Let Assumption A.1 hold and . For every , there exist constants , independent of the discretized parameters , and such that for all sufficiently small, there holds
The proof is detailed in Sections 3, 4 and 5 and uses the semi-linear structure of along with the weak monotonicity property (1.3). We first consider a semi-discrete (in time) scheme (4.1) of problem (1.2) and derive the error estimate between the strong solution X of (1.2) and the discretized solution of (4.1), see Theorem 4.2. Again, using uniform bounds for higher moments of the solutions and , we derive the error estimate of and in strong norm, cf. Theorem 5.2. Putting things together then settles the main theorem.
3 Stochastic Allen–Cahn Equation: The Continuous Case
In this section, we derive uniform bounds of arbitrary moments for the strong solution of (1.2) and using these uniform bounds, we estimate the expectation of the increment in terms of .
The following estimate may be shown by a standard Galerkin method which employs a (finite) sequence of (-orthonormal) eigenfunctions of the inversely compact, self-adjoint isomorphic operator , the use of Itô’s formula to the functional , and the final passage to the limit (see e.g. ),
for which we require the improved regularity property . Thanks to Assumption A.1, we see that σ satisfies the following estimates:
There exists a constant such that
There exist constants and such that for all ,
Let Assumption A.1 hold and . Then there exists a constant such that
Suppose in addition that . Then there exists a constant such that
(i) We proceed formally. Note that
We use Itô’s formula for :
where for all . By the Cauchy–Schwarz inequality, and (3.2), we have
Since σ is bounded, by using Young’s inequality, we have
We choose such that . With this choice of θ, by (3.1), we have for some constant ,
We use Gronwall’s lemma to conclude that
(ii) Use Itô’s formula for . We compute its derivatives:
Note that by integration by parts
Because of for we estimate the last term through
Note that , and therefore we see that
Let . Then by (3.3), we see that
Let . Then, thanks to (3.3) and the Cauchy–Schwarz inequality, we have
Let Assumption A.1 hold and . Then, for every , there exists a constant such that
Fix . An application of Itô’s formula for with to (1.2) yields, after taking expectation
We use the weak monotonicity property (1.3) to bound from below the second term on the left-hand side,
The integration by parts formula and Young’s inequality reveal that
Since for , by using Young’s inequality, we see that
Again, thanks to the boundedness of σ and Young’s inequality, we see that
(ii) We apply Itô’s formula to the function for any to (1.2), and then use for fixed and integrate with respect to spatial variable. Thanks to Young’s inequality, and the boundedness of ,
4 Semi-Discrete Scheme (in Time) and Its Bound
Let be an equi-distant partition of of size . The structure preserving time discrete version of (1.2) defines an -adapted -valued process such that -almost surely and for all ,
where and f are defined in (2.6). Solvability for easily follows from a coercivity property of the drift operator, and the Lipschitz continuity property (2.1) for the diffusion operator. Below, we denote again
The proof of the following lemma evidences why is substituted by in (4.1) to recover uniform bounds for arbitrary higher moments of .
Suppose that , and that Assumption A.1 holds. For every , , there exists a constant such that
(1) Consider (4.1) for a fixed and choose . Then one has -a.s.,
By using the identity for all along with integration by parts formula, we calculate
Since is bounded, we observe that
We decompose into the sum of two terms and , where
In view of Young’s inequality and the boundedness of σ, we have
Next we estimate independently to bound the term . To do so, we choose as test function in (4.1) and obtain
Again, since is a bounded domain, one has
Summation over all time steps, and the discrete Gronwall’s lemma then establish the assertion for .
(2) In order to validate the assertion for , we proceed inductively and illustrate the argument for . Recall that we have from before
To prove the assertion for , one needs to multiply (4.8) by some quantity to produce a term like with on the left-hand side of the inequality in order to absorb related terms coming from the right-hand side of the inequality before discrete Gronwall’s lemma. Therefore, we multiply (4.8) with to get by binomial formula
By Young’s inequality, we have
We can decompose as
Note that . By using Young’s inequality and the boundedness of , we estimate ,
Again, can be written as with , where
Thanks to Young’s inequality, the boundedness of σ and (4.7) we get for ,
We combine all the above estimates in (4.9), and choose with to have, after taking expectation
Summation over all time steps in (4.10), together with the discrete Gronwall’s lemma then validates the assertion of the theorem for . This completes the proof. ∎
Assume that , and Assumption A.1 holds true. Then, for every , there exist constants and such that for all sufficiently small,
Consider (2.2) for the time interval , and denote . There holds -a.s. for all ,
The third term on the right-hand side attributes to the use of instead of in (4.1). Choose , and apply expectation. By the weak monotonicity property (1.3) of the drift, the left-hand side of (4.11) is then bounded from below by
Because of Young’s inequality and Lemma 3.2 (ii), we conclude
Next we bound . For this purpose, we use the embedding for , the algebraic identity , and Young’s and Hölder’s inequalities in combination with Lemma 3.2 to estimate ()
It is immediate to validate
by adding and subtracting in the second argument and proceeding as before, and Itô’s isometry in combination with (2.1) and Lemma 3.2 (i). Next we focus on the term . In view of generalized Hölder’s inequality, and the embedding for ,
In view of Lemma 4.1, we see that
and therefore we obtain
We combine all the above estimates to have
Summation over all time steps in (4.14), together with the discrete (implicit form) Gronwall’s lemma then validates the assertion of the theorem. ∎
5 Space-Time Discretization and Strong Error Estimate
In this section, we first derive the uniform moment estimate for the discretized solution of the structure preserving finite element based fully discrete scheme (2.5). Then by using these uniform bounds along with Lemma 4.1 we bound the error , where solves (4.1).
We define the discrete Laplacian by the variational identity
One can use the test function in (2.5) and proceed as in the proof of Lemma 4.1 along with (2.4), the - and -stabilities of the projection operator (cf. ) to arrive at the following uniform moment estimates for .
For every , , there exists a constant such that
In view of Lemma 5.1, it follows that
We have the following theorem regarding the error in strong norm.
Assume that . Then, under Assumption A.1, there exist constants , independent of the discretization parameters , and such that for all sufficiently small, there holds
Note that the third term on the right-hand side of the first equality reflects that is a perturbation of . By the weak monotonicity property (1.3), we see that
and therefore we arrive at the following inequality
Note that, in view of Lemma 4.1, Young’s inequality and the embedding for
Thus using Lemma 4.1 and the estimate above, we see that
Let us recall the following well-known properties of , see :
We now bound the term . We use the algebraic formula given before (4.12), the embedding for , and a generalized Young’s inequality to have
It remains to bound . We decompose as follows:
Next we estimate . We use the generalized Hölder’s inequality, the -stability of , the embedding for , Young’s inequality, estimates (5.1), (5.4) and (4.6), (4.7) and (4.13), along with Lemma 4.1 to get
Putting things together in (5.2) and using the discrete Gronwall’s lemma (implicit form) then yields
This finishes the proof. ∎
5.1 Proof of Main Theorem
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About the article
Published Online: 2017-07-18
Published in Print: 2018-04-01