An -Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems
In this paper we develop an -adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an -version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully -adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.
Key words and phrases:Adaptive Newton methods, semilinear elliptic problems, singularly perturbed problems, adaptive finite element methods, -FEM, -adaptivity.
2010 Mathematics Subject Classification:49M15,58C15,65N30
The purpose of this paper is the development of a numerical approximation procedure for semilinear elliptic boundary value problems, with possible singular perturbations. More precisely, for a fixed parameter (possibly with ), and a continuously differentiable function , we consider the problem of finding a function in an interval , , which satisfies
Solutions of (1.1) are typically not unique (even infinitely many solutions may exist), and, in the singularly perturbed case, may exhibit boundary layers, interior shocks, and (multiple) spikes. The existence of multiple solutions due to the nonlinearity of the problem and/or the appearance of singular effects constitute two challenging issues when solving problems of this type numerically; see, e.g.,[23, 27].
Newton-Galerkin Methods: In order to address the numerical solution of nonlinear singularly perturbed problems of type (1.1), a Newton-Galerkin approach has recently been proposed in [2, 3]. It is built upon two main ideas: Firstly, adaptive Newton methods are applied on the continuous level in order to approximate the nonlinear differential equation (1.1) by a sequence of linearized problems. Secondly, the resulting linear problems are discretized by means of an adaptive finite element procedure which, in turn, is based on an -robust a posteriori error analysis. Then, providing an appropriate interplay between the adaptive Newton method and the adaptive finite element approach, by which we either perform a Newton step (if the Newton linearization effect dominates) or refine the current finite element mesh based on some suitable a posteriori error indicators (in case that the finite element discretization causes the main source of error), a fully adaptive Newton-Galerkin scheme is obtained; see also the related works on monotone nonlinear problems [4, 9, 6], or the articles [5, 13] dealing with modelling errors in linearized models. Our numerical results in  reveal that sensible solutions can be found even in the singularly perturbed case, and that our scheme is reliable in the sense that the adaptive Newton-Galerkin method is able to converge to solutions which preserve the qualitative structure of (reasonably chosen) initial guesses (see also [2, 24]).
-FEM and -adaptivity: The aim of the present paper is to extend our work in  from a low-order to an -adaptive Newton-Galerkin approach. For simplicity of exposition, we focus on the 1d case, however, we mention that an extension to the multi-d case is possible as well. Exploiting the -framework may potentially lead to exponential rates of convergence in the numerical approximation. For the purpose of -adaptivity, we carry out an -robust -version a posteriori residual analysis for the FEM discretizations of the Newton linearizations. We remark that using -adaptivity, in contrast to the adaptive low-order methodology developed in the previous works [9, 2, 3], introduces an additional aspect of adaptivity that may further improve the performance of the Newton-Galerkin approach. In the -literature, several adaptive strategies and algorithms have been proposed; see, e.g., the overview article  and the references therein. In particular, we point to the smoothness estimation techniques proposed in [8, 15, 16, 18], and the related approach [11, 12, 28] involving continuous Sobolev embeddings. The latter works will be applied in the current paper in order to drive the -adaptive finite element refinements; in combination with the adaptive Newton linearizations this will lead to a fully -adaptive Newton-Galerkin procedure.
Problem formulation and notation: Henceforth, we suppose that a (not necessarily unique) solution of (1.1) exists; here, we denote by the standard Sobolev space of functions in with zero trace on . Furthermore, signifying by the dual space of , and upon defining the map through
where is the dual product in , the above problem (1.1) can be written as a nonlinear operator equation in :
In addition, for open subsets , we introduce the norm
where denotes the -norm on . Note that, in the case of , with given , i.e., when (1.1) is linear and strongly elliptic, the norm is a natural energy norm on . Frequently, for , the subindex ‘’ will be omitted. Furthermore, the associated dual norm of from (1.2) is given by
Throughout this work we shall use the abbreviation to mean , for a constant independent of the mesh size , the local polynomial degree , and of .
Outline: In Section 2 we will consider the Newton method within the context of dynamical systems in general Banach spaces, and present a prediction strategy for controlling the Newton step size parameter. Furthermore, Section 3 focuses on the application of the Newton methodology to semilinear elliptic problems as well as on the discretization of the problems under consideration by -finite element methods; in addition, we derive an -robust a posteriori residual analysis. An -adaptive procedure and a series of numerical experiments illustrating the performance of the proposed idea will be presented in Section 5. Finally, we summarize our findings in Section 6.
2. Adaptive Newton Methods in Banach Spaces
In the following section we shall briefly revisit an adaptive prediction-type Newton algorithm from .
2.1. Abstract Framework
Let be two Banach spaces, with norms and , respectively. Given an open subset , and a (possibly nonlinear) operator , we consider the nonlinear operator equation
for some unknown zeros . Supposing that the Fréchet derivative of exists in (or in a suitable subset), the classical Newton method for solving (2.1) starts from an initial guess , and generates the iterates
where the update is implicitly given by the linear equation
Naturally, for this iteration to be well-defined, we need to assume that is invertible for all , and that .
2.2. An Adaptive Prediction Strategy
In order to improve the reliability of the Newton method (2.2) in the case that the initial guess is relatively far away from a root of , , introducing some damping in the Newton method is a well-known remedy. In that case (2.2) is rewritten as
where , , is a damping parameter that may be adjusted adaptively in each iteration step.
Provided that is invertible on a suitable subset of , we define the Newton transform by
Then, rearranging terms in (2.3), we notice that
i.e., (2.3) can be seen as the discretization of the Davydenko-type system,
by the forward Euler scheme, with step size .
For , the solution of (2.4), if it exists, defines a trajectory in that starts at , and that will potentially converge to a zero of as . Indeed, this can be seen (formally) from the integral form of (2.4), that is,
which implies that as .
Now taking the view of dynamical systems, our goal is to compute an upper bound for the value of the step sizes from (2.3), , so that the discrete forward Euler solution from (2.3) stays reasonably close to the continuous solution of (2.4). Specifically, for a prescribed tolerance , a Taylor expansion analysis (see [3, Section 2] for details) reveals that
where, for any sufficiently small , we let . Hence, after the first time step of length there holds
where is the forward Euler solution from (2.3). Therefore, upon setting
we arrive at
In order to balance the -terms in (2.5) it is sensible to make the choice
for some parameter . This leads to the following adaptive Newton algorithm.
Fix a tolerance as well as a parameter , and set .
The following comments are noteworthy.
We remark that the purpose of the above derivations is to work under minimal structural assumptions on the dynamics caused by the Newton transform. This is particularly important in the context of nonlinear singularly perturbed problems (with ): Indeed, for such problems (as ) the inverse of the derivative of the associated nonlinear operator will typically become highly irregular (even on a local level), and, for instance, local or global Lipschitz type or boundedness assumptions (involving, e.g., second derivative information) will in general not be available with practically feasible constants. We remark, however, that more sophisticated step size prediction schemes can be derived if the structure of the nonlinearities can be exploited further; see, in particular, the monograph .
The minimum in (2.7) ensures that the step size is chosen to be 1 whenever possible. Indeed, this is required in order to guarantee quadratic convergence of the Newton iteration close to a root (provided that the root is simple).
The preset tolerance in the above adaptive strategy will typically be fixed a priori. Here, for highly nonlinear problems featuring numerous or even infinitely many solutions, it is typically mandatory to select small in order to remain within the attractor of the given initial guess. This is particularly important if the starting value is relatively far away from a solution.
3. Newton-Galerkin -FEM for Semilinear Elliptic Problems
The purpose of this section is to derive an -version Newton-Galerkin finite element formulation for the solution of (1.1). To this end, we will apply the abstract adaptive Newton framework from the previous Section 2, and apply an -discretization methodology for the resulting sequence of (locally) linearized problems.
3.1. Newton Linearization
where we write . We note that, if , then is a well-defined linear and bounded mapping from to ; see [3, Lemma A.1].
in . Equivalently,
where, for fixed ,
are bilinear and linear forms on and , respectively.
Incidentally, if there are constants with such that holds for all , where is the (best) constant in the Poincaré inequality on , i.e.,
see [14, Theorem 257], then, it is elementary to show that the formulation (3.1) is a linear second-order diffusion-reaction problem, which is coercive on . In consequence, for any given , it has a unique solution ; see  for details.
3.2. Linearized Finite Element Approach
In order to provide a numerical approximation of (1.1), we will discretize the weak formulation (3.1) by means of an -finite element method, which, in combination with the Newton iteration, constitutes a Newton-Galerkin approximation scheme. Furthermore, in view of deriving an -adaptive algorithm, we will develop an -robust -version a posteriori residual analysis for the linear finite element formulation.
We introduce a partition of (open) elements , , on , with . The length of an element is denoted by , . For each element , it will be convenient to introduce the patch as the union of and of the elements adjacent to it. In addition, to each element we associate a polynomial degree , . These numbers are stored in a polynomial degree vector . Then, we define an -finite element space by
where, for , we denote by the space of all polynomials of degree at most . We say that the pair of a partition and of a degree vector is -shape regular, for some constant independent of , if
i.e., if both the element sizes and polynomial degrees of neighboring elements are comparable.
3.2.2. Linear -Fem
We define the numerical approximation of (1.1) in an iterative way as follows: For , we let be an initial -discretization space, and a suitable initial guess for the Newton iteration. Furthermore, for , and given , we find a finite element approximation of (3.1) as the solution of the weak formulation
Here, takes the role of a (possibly varying) step size parameter in the adaptive Newton scheme as described earlier in Section 2.2.
4. A Posteriori Residual Analysis
Recalling (3.4), for given , let us introduce the function
as well as
Then, rearranging terms, we can rewrite (3.4) in the following form:
The aim of this section is to derive a posteriori residual-based bounds for (4.3). In order to measure the discrepancy between the finite element discretization (3.4) and the original problem (1.1), an obvious quantity to bound is the residual in . In order to proceed in this direction, we notice that the adaptively chosen damping parameter in the Newton method (3.4) will equal 1 sufficiently close to a root of . For this reason, as was proposed in , we may focus on the ‘shifted’ residual in instead; indeed, estimating this quantity turns out to be very natural in view of the linearized finite element discretization (4.3).
4.1. Robust Interpolation Estimates
For the purpose of deriving an (upper) a posteriori residual estimate that is robust with respect to the singular perturbation parameter (for ) as well as optimally scaled with respect to the local element sizes and polynomial degrees , it is crucial to construct an interpolation operator that is simultaneously - and -stable. This will be accomplished in the current section; see Proposition 4.1 and Corollary 4.2 below.
Let the pair be -shape regular; see (3.3). Then, there exists an interpolation operator such that, for any , and for all , there holds
Here, is a constant that depends solely on ; in particular, it is independent of , , and of .
Let us, without loss of generality, assume that . The result can be shown with the techniques employed in the higher-dimensional case in [17, Cor. 3.7]. In the present, one-dimensional case, a significantly simpler argument can be brought to bear, which allows us to make the paper self-contained. Let and and , be the standard piecewise linear hat functions associated with the nodes , . The extra nodes and define in a natural way the elements and . The (open) patches , , are given by the supports of the functions , i.e., .
Polynomial approximation (see, e.g., [19, Proposition A.2]) gives the existence of an interpolation operator that is uniformly (in ) stable, i.e., for all and has the following properties for :
Furthermore, if is antisymmetric with respect to the midpoint , then can be assumed to be antisymmetric as well, i.e., (this follows from studying the antisymmetric part of the original function ).
The approximation is now constructed with the aid of a “partition of unity argument” as described in [20, Theorem 2.1]. For and , extend anti-symmetrically, i.e., for and for . Then is defined on each patch , . For each patch , let (with the understanding and ). The above operator then induces for each patch by scaling an operator with the following properties:
here, we have exploited the -shape regularity of the mesh. We note that and . Also, the operators are uniformly (in the polynomial degree) stable in . The approximation is now taken to be . The desired approximation properties follow now from [20, Theorem 2.1]. ∎
The above proposition implies the following bounds.
Combining these two estimates, yields the first bound.
In order to prove the second estimate, we apply, for , a multiplicative trace inequality; see [21, Lemma A.1]):
Then, invoking the above bounds as well as the estimates from Proposition 4.1, we get
with from (4.6). Since is also a boundary point of , we similarly obtain that
with from (4.7). Thus, we have shown the second estimate. ∎
4.2. Robust A Posteriori Residual Estimate
Based on the previous interpolation analysis in Section 4.1, we are now in a position to prove the following main result.
We emphasize that the constants and appearing in the error indicators from (4.9) remain bounded as (and ). In addition, we note that , and that .
For linear problems, the (adaptive) Newton linearization is redundant. In particular, upon setting , we infer that in (4.8), and the residual is bounded by the local error indicators only. These quantities, in turn, can be exploited for the purpose of designing an -adaptive mesh refinement strategy, as outlined, for example, in Section 5.1 below. We emphasize that this approach has been studied earlier in the report , where a number of numerical tests in the context of -adaptive FEM for singularly perturbed linear problems have been performed; in particular, these experiments have underlined the robustness of the bound (4.8) as in the linear case. In Section 5, we will extend this idea by proposing a suitable interplay between -mesh refinements and the (adaptive) Newton linearization from Algorithm 2.1.
Proof of Theorem 4.3.
see [3, Proposition 4.3].
Applying the Cauchy-Schwarz inequality leads to
Furthermore, again using Corollary 4.2, we see that
Similarly, there holds
Now, applying the Cauchy-Schwarz inequality to (4.11) we see that
Dividing by , and taking the supremum for all , completes the proof. ∎
Following along the lines of [3, §4.4.2] and [21, Appendix], it is possible to prove -robust local lower residual bounds in terms of the error indicators and the data oscillation terms. This approach, however, results in local efficiency bounds that will be slightly suboptimal with respect to the local polynomial degrees due to the need of applying -dependent norm equivalences (involving cut-off functions).
5. An -Adaptive Newton-Galerkin Procedure
In this section, we will discuss how the a posteriori bound from Theorem 4.3 can be exploited for the purpose of designing an -adaptive Newton-Galerkin algorithm for the numerical solution of (1.1). We will begin by revisiting an -adaptive testing strategy from  (see also [28, 12]).
5.1. -Adaptive Refinement Procedure
In order to -refine the -finite element space , we shall apply an -adaptive algorithm which is based on the following two ingredients.
(a) Element marking:
The elementwise error indicators from Theorem 4.3 are employed in order to mark elements for refinement. More precisely, we fix a parameter , and select elements to be refined according to the Dörfler marking criterion:
Here, the indices are chosen such that the error indicators from (4.9) are sorted in descending order, and is minimal.
(b) -refinement criterion:
The decision of whether a marked element in step (a) is refined with respect to (element bisection) or (increasing the local polynomial order by 1) is based on a smoothness testing approach. Specifically, if the (numerical) solution is considered smooth on a marked element , then the polynomial degree is increased by 1 on that particular element (no element bisection), otherwise the element is bisected (retaining the current polynomial degree on both subelements). In order to evaluate the smoothness of the solution on a marked element , we employ an elementwise smoothness indicator as introduced in [11, Eq. (3)],