Abstract
In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.
1 Introduction
Adaptive mesh-refining plays an important practical role in accurate and efficient calculation of the numerical solutions of partial differential equations.
A posteriori error estimates are an essential ingredient of adaptivity.
They are composed of the known values such as the given data of the problem and the computed numerical solutions.
Various a posteriori error estimates for finite element methods for second-order elliptic problems have already been studied in [1, 2, 8, 17, 19], for the Stokes problem in [9, 26], and for the Navier–Stokes problem in [12, 14].
Particularly, the nonlinear elliptic equations were considered in [19].
There, a priori and a posteriori error estimates measured in the
In this paper, we derive a unified a posteriori error estimate for the approximate solutions of the multi-level algorithm developed in [22] for a class of nonlinear problems in the framework of BRR. The multi-level algorithm using a two-grid idea is an efficient numerical method designed to resolve nonlinearities. The two-grid algorithm (see [20, 27, 28]) is first applied, and in the subsequent process, uniform mesh refinement is exploited to deliver a desired accuracy and convergence; as the mesh is being refined, the solution on a given mesh is exploited as an accurate starting iterate for solutions on the next mesh level. The multi-level algorithm has quadratic convergence in the sense of equation (2.5) in Theorem 2.5. We apply this strategy to efficiently compute numerical solutions by adaptive mesh refinement. We emphasize that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is validated from the numerical results presented in the last section. The BRR framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the nonlinear problem. We mention that this approach is quite general and can be applied to any numerical scheme for nonlinear problems satisfying the BRR framework.
As applications, we consider the pseudostress-velocity formulation of the stationary Navier–Stokes equations (NSE) and the standard Galerkin formulation of semilinear elliptic equations (SEE).
Finite element methods for the velocity-pressure and the stress-velocity-pressure formulation of the NSE have been analyzed for decades (cf. [3, 13]).
The stress-velocity-pressure formulation is the original physical equations for incompressible Newtonian flows induced by the conservation of momentum and the constitutive law.
The mixed finite element methods for the pseudostress-velocity formulation have been recently established in for the Stokes [6, 10] and the Navier–Stokes equations [5, 21].
The pseudostress-velocity formulation has several advantages.
The pseudostress is nonsymmetric unlike stress tensor and approximated by the
For adaptivity, we derive reliable and efficient residual-based a posteriori error estimators for our multi-level algorithm for the NSE. We mention existing works on the adaptive two-level algorithm. The authors in [12, 14] derived reliable a posteriori error estimators for the two-level methods for the velocity-pressure formulation of the NSE. In comparison to usual adaptive finite element methods [10, 18, 19, 23, 25], the estimators for two-level algorithm contain an additional term which measures the difference between coarse and fine grid solutions. If the additional term measures the difference between consecutive levels and is of higher order (and negligible), then the link between levels does not effect the computation, which is computationally more advantageous. Indeed, this additional term is negligible in our multi-level setting in contrast to [14]. See Remark 3.19 for further details. As the second application of the unified framework, we consider the standard Galerkin method for the SEE and derive reliable a posteriori error estimators. It turns out that behavior of the additional term arising from SEE is similar to that of NSE. Note that our algorithm can be applied to other problems such as the elliptic problem with gradient nonlinearities (cf. [24, 16, 19, 20]).
The remainder of this paper is organized as follows. In the next section, we introduce approximation of branches of nonsingular solutions based on the BRR theory and multi-level algorithm analyzed in [22]. We derive a unified a posteriori error estimate for the approximate solutions of the multi-level algorithm for a class of nonlinear problems in the framework of BRR. Section 3 and Section 4 are devoted to applying the algorithms to the NSE and the SEE, respectively, and constructing a posteriori error estimators of residual type. Section 5 presents our adaptive multi-level algorithm. In Section 6, numerical results are presented to show the performance of the algorithms and to validate the theory developed in this paper. Concluding remarks are given in the last section.
2 BRR Theory and Multi-level Algorithm
In this section, we start with the abstract approximation theory of Brezzi, Rappaz and Raviart for nonlinear problems developed in [4, 7, 13]. Then we introduce the multi-level algorithm analyzed in [22], which is a method to solve resulting nonlinear algebraic equations.
From here to the Remark 2.6, we will use the same parts of what we described in [22].
Let 𝑋, 𝒳 and 𝑌 be real Banach spaces with the norms
where
We assume that the following hypotheses.
𝐺 is a
There exists another Banach space
For all
Assume that (H1)–(H4) hold and that
Note that, since Λ is compact, the function
Assume that (H1)–(H4) hold and that
When 𝐹 is replaced by
We introduce the multi-level version of the two-grid algorithm.
For
Algorithm 1 (Multi-level algorithm)
(Nonlinear solver) Solve nonlinear system on initial coarse mesh.
Find
(Linear solver) Update on each finer mesh level 𝑗 with one Newton iteration.
For
The two-grid scheme in [27, 28] is based on passing information between finite element equations defined on two grids of different mesh sizes.
In the first step, a nonlinear problem itself is solved in a coarse space, i.e., a finite-dimensional space with coarse grids.
In the second step, the nonlinear problem is linearized locally at the solution obtained in the coarse space.
Then the linearized problem is solved in a fine space.
For better performance, this process can be iterated on a sequence of linearized problems with increasing dimensions.
In the second step of the multi-level algorithm in [14], the nonlinear problem is linearized locally at the solution
Theorem 2.5 (A Priori Error Bound)
Assume that (H1)–(H4) hold.
Let
and for
It should be noted that the above algorithm can be applied even if
Now, we want to find an a posteriori error bound for solutions obtained from the multi-level algorithm.
First, we introduce the following theorem presented and proved in [7, Theorem 2.1].
Let
We assume that
Moreover, for all
Theorem 2.8 (A Posteriori Error Bound)
Assume that (H1)–(H4) hold.
Let
Proof
The proof follows by applying Theorem 2.7.
For this, we first notice that
To apply Theorem 2.7, we set
From the mean value theorem and (H1), we have, for
which yields that there exist
Applying Theorem 2.5, the mean value theorem and (H1) to the identity
we can find
Applying Theorem 2.7 with
From the uniqueness of 𝜓 in
3 Application 1: Navier–Stokes Equations
In this section, we introduce the a priori error estimate and analyze an a posteriori error for multi-level algorithms for the pseudostress-velocity formulation of the NSE.
The a priori error was analyzed in [21].
The following details are only given for the reader’s convenience.
First, we introduce some notation and function spaces.
Let
Given data 𝒇 and 𝒈, the stationary and incompressible NSE for the unknown velocity 𝒖 and the pressure 𝑝 reads
where 𝒏 is the outward unit normal vector to the boundary.
Let
The pseudostress tensor
The compatibility condition
Scaling pseudostress 𝝈 and data 𝒇 by
Assume that Ω is a bounded, open, connected subset of
We define spaces
Then the mixed variational problem of the pseudostress-velocity formulation (3.6)–(3.7) is to find
such that
Let 𝒇 be in
3.1 BRR Framework for NSE
We let
for
for
Here we used the triangle inequality and the Hölder inequality.
So hypothesis (H2) holds.
From the definition, 𝐺 belongs to the
We let
defined by the solution of
For any
Clearly, the pair
if and only if the function
For any
Proof
See [21]. ∎
We now consider mixed finite element approximations.
Let
where
for
For short notation on generic constants 𝐶 independent of the local or global mesh sizes, for any expressions 𝐴 and 𝐵,
We introduce a projection operator over the space 𝑊, where
where
But, in general, the above relation does not hold for the operator
Let
Then the following lemma holds (see [6] for details).
The commutative property
The mixed finite element method for approximation of the solution of system (3.8)–(3.9) is defined by finding
Next, we define a discrete solution operator
where
Note that
For any
Moreover, assume that
Proof
See [21]. ∎
Theorem 3.5 implies that hypotheses (H3) and (H4) hold. We get the following theorem.
Let
Moreover, assume that
where
Proof
See [21]. ∎
3.2 Error Estimates for Multi-level Algorithm for NSE
Now, we apply the multi-level algorithm to NSE based on pseudostress-velocity formulation.
Note that the Fréchet derivative of 𝐹 at
for any
We generate a shape-regular triangulation
Algorithm 2 (Multi-level algorithm for NSE)
(Nonlinear solver) Find
(3.21)(3.22)(Linear solver) For
(3.23)(3.24)
If
Theorem 3.7 (A Priori Error Estimates)
Let
Proof
See [21]. ∎
Theorem 2.8 indeed provides the reliability bound.
For brevity, 𝜈 is dropped from the functions
where
Equation (3.25) says that an a posteriori error estimate for the NSE is controlled by that for the Stokes equations with the additional term appearing in the right-hand side of (3.28). Thus, we can apply existing a posteriori error estimates for the pseudostress-velocity formulation of the Stokes equations to the NSE. Indeed, we apply the residual-based a posteriori error estimates for the two-dimensional Stokes equations analyzed by Carstensen, Kim and Park in [10].
Let us describe the well-known results used throughout this paper.
Assume that Ω is a bounded Lipschitz domain with a polygonal boundary.
Then there exists a constant
The general interpolation operator cannot be applied to
Note that, since the triangulation
Finally, we state the usual inverse inequalities introduced in [25] which we will use in order to derive local lower bounds for the error. We need some modified results whose proof easily follows from the idea of the paper [25], and we omit the proof.
There exist constants
Let
For the error
where
Proof
Since
Then, by integration by parts, (3.26) and the relation
Equations (3.23), (3.37), (3.27) and (3.24) imply
The last term of (3.39) is derived by applying the compatibility condition
Applying estimates (3.14), (3.15) and (3.38) to the above equation (3.39), we conclude the proof. ∎
To derive the estimate of the error bound for
When
We notice that, for
where
For the error
with the estimate
where 𝐶 is a positive constant independent of 𝜺.
Proof
We consider the following Stokes problem: find
The solutions 𝒛 and
Hence
where
We define the jump
and
For the error
Proof
In the Helmholtz decomposition (3.41) of
we choose 𝒓 satisfying
From the above decomposition and the identity
Thus
From (3.43), we have the error decomposition
Integration by parts, (3.24) and (3.27) imply
Using inequality (3.14), Cauchy–Schwarz inequality and Sobolev imbedding theorem, we have
We now consider the second term
We find, from (3.23) and (3.26),
Moreover, from the fact that
Considering integration by parts (3.40) and estimate (3.30)–(3.31), recalling that the number of elements in
Putting (3.45) and (3.46) into (3.44) and using estimate (3.42), we conclude the proof. ∎
Combining Corollary 2.8 and Lemmas 3.11–3.12, we obtain the following reliability estimate.
Theorem 3.13 (Reliable A Posteriori Error Estimates)
For the solution
where the error estimator
We note that physical quantities such as pressure, gradient velocity and stress can be expressed as follows:
From these identities, the approximation of pressure, gradient velocity and stress can be defined by
Then the following relations hold:
Now, we derive an efficient a posteriori error estimate.
Let
For each
Proof
Using inverse inequality (3.32), the identity
By the Hölder inequality, inverse inequality (3.32) and (3.35), and the mean value theorem, we get
With an application of the triangle inequality to the identity
we complete the proof by (3.47). ∎
Since
For each
Proof
By inequality (3.32) and integration by parts, we have
Applying inverse inequality (3.35),
Putting (3.49) into (3.48) and using the relation
For each
Proof
Inequality (3.32), the identity
We obtain, from inverse inequality (3.35),
Hence, combining (3.50) and (3.51), we conclude the proof. ∎
For each
where
Proof
Set
From Lemma 3.17, we have
Application of (3.34), (3.36) and (3.53) to (3.52) yield
where
By the Hölder inequality and the triangle inequality, we have
Assume that
Similarly,
Hence
which shows the asymptotic behavior of the additional term. Note that the additional term
is of higher order compared to the other error estimates and the error
Combining Lemmas 3.15–3.18, we arrive at the following efficiency estimates.
Theorem 3.20 (Efficient A Posteriori Error Estimate)
For the solution
where
Note that the derived a posteriori error estimator is efficient and reliable, i.e., the efficiency index
and its reciprocal are bounded for all triangulations.
4 Application 2: Semilinear Elliptic Equations
We consider the semilinear boundary value problems of the form
where
4.1 BRR Framework for SEE
We define the continuous linear operator
Introducing the operator 𝑆, problem (4.2) can be written in the equivalent way
In order to approximate the solution of problem (4.3), given a finite-dimensional space
Then the approximation of nonlinear problem (4.3) is to find
We analyze problem (4.1) with
where
4.2 Error Estimates for Multi-level Algorithm for SEE
We generate a shape-regular triangulation
Algorithm 3 (Multi-level algorithm for SEE)
(Nonlinear solver) Find
(4.6)(Linear solver) For
(4.7)
Theorem 4.1 (A Priori Error Estimates)
Let
By referring to [20], we can prove that the result of (2.5) holds for
which shows improvement
From Theorem 2.8, we observe that
where
The following error equations immediately follow from (4.7)–(4.9).
For
Let
Theorem 4.3 (Reliable A Posteriori Error Estimates)
For the error
and
Proof
By integration by parts and (4.10), we get
where
and for
Assume that
and
where a constant 𝐶 depends on
5 Adaptive Multi-level Algorithm
We introduce an adaptive multi-level algorithm for nonlinear problems based on the local error estimator
The step SOLVE means to perform Step 2 of Algorithm 1 on the mesh
Algorithm 4 (Adaptive multi-level algorithm)
Multi-level process computing
For the reader’s convenience, we present Figure 1 in order to explain the outline of the (uniform/adaptive) multi-level algorithm.
The details of the uniform multi-level algorithm can be found in [22].
Figure 1 depicts the multi-level algorithm compared with the usual nonlinear solver (e.g., Newton–Raphson algorithm).
The nonlinear solver finds discrete nonlinear solutions by iteration on a much higher-dimensional fixed space
6 Numerical Experiments
In this section, we perform various numerical experiments to test our adaptive multi-level algorithm and to validate the theory analyzed in this paper. We present the convergence of various errors and indicators on both uniform and adaptive meshes. In particular, we will numerically verify that the adaptive multi-level algorithm has the quadratic convergence in the sense of (2.5).
6.1 Semilinear Elliptic Equations
We approximate the semilinear elliptic problem (4.1) with
We define the local error estimator
the global error estimator
and efficiency indices
To show the performance, we consider two test examples: one with a boundary layer, the other with a corner singularity.
Example 6.1 (A Solution with a Boundary Layer)
We consider SEE (4.1) with
The exact solution 𝑢,
specifies data 𝑓.
In Table 1, we present convergence history of multi-level algorithm and nonlinear solver on the uniform meshes.
Here,
In Table 2 and Figures 2 and 3, we present numerical results from the adaptive multi-level algorithm.
Efficiency indices
In Figure 3, we observe that the additional term
Convergence orders (CO) of various errors on uniform meshes for Example 6.1
| Multi-level algorithm | Nonlinear solver | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
|
|
||||||||||
| 𝛼 | 𝑗 |
|
|
CO |
|
CO |
|
CO |
|
CO |
| 2 | 0 | 256(8) | 6.6901e−1 | 1.9446e+1 | 6.6901e−1 | 1.9446e+1 | ||||
| 1 | 1024(16) | 2.0875e−1 | 1.68 | 1.1322e+1 | 0.78 | 2.0263e−1 | 1.72 | 1.1314e+1 | 0.78 | |
| 2 | 4096(32) | 5.3730e−2 | 1.96 | 5.8497e+0 | 0.95 | 5.3255e−2 | 1.93 | 5.8494e+0 | 0.95 | |
| 3 | 16384(64) | 1.3509e−2 | 1.99 | 2.9470e+0 | 0.99 | 1.3482e−2 | 1.98 | 2.9470e+0 | 0.99 | |
| 4 | 65536(128) | 3.3828e−3 | 2.00 | 1.4762e+0 | 1.00 | 3.3812e−3 | 2.00 | 1.4762e+0 | 1.00 | |
| 5 | 262144(256) | 8.4606e−4 | 2.00 | 7.3843e−1 | 1.00 | 8.4596e−4 | 2.00 | 7.3843e−1 | 1.00 | |
|
|
||||||||||
| 3 | 0 | 256(8) | 6.1077e−1 | 1.9426e+1 | 6.1077e−1 | 1.9426e+1 | ||||
| 1 | 1024(16) | 2.1849e−1 | 1.48 | 1.1368e+1 | 0.77 | 1.8097e−1 | 1.75 | 1.1307e+1 | 0.78 | |
| 2 | 4096(32) | 5.1247e−2 | 2.09 | 5.8541e+0 | 0.96 | 4.7263e−2 | 1.94 | 5.8500e+0 | 0.95 | |
| 3 | 16384(64) | 1.2075e−2 | 2.09 | 2.9472e+0 | 0.99 | 1.1936e−2 | 1.99 | 2.9471e+0 | 0.99 | |
| 4 | 65536(128) | 2.9973e−3 | 2.01 | 1.4762e+0 | 1.00 | 2.9917e−3 | 2.00 | 1.4762e+0 | 1.00 | |
| 5 | 262144(256) | 7.4873e−4 | 2.00 | 7.3844e−1 | 1.00 | 7.4840e−4 | 2.00 | 7.3844e−1 | 1.00 | |
Efficiency indices on adaptive meshes for Example 6.1
|
|
|
||||||||
|---|---|---|---|---|---|---|---|---|---|
|
|
|||||||||
| 𝑗 |
|
|
|
|
𝑗 |
|
|
|
|
| 0 | 256 | 1.9446e+1 | 0 | 256 | 1.9426e+1 | ||||
| 1 | 1024 | 1.1320e+1 | 5.7659e+1 | 5.09 | 1 | 1024 | 1.1368e+1 | 5.8250e+1 | 5.12 |
| 2 | 1076 | 7.5222e+0 | 3.8374e+1 | 5.10 | 2 | 1076 | 7.5233e+0 | 3.8552e+1 | 5.12 |
| 3 | 1260 | 5.1540e+0 | 2.5581e+1 | 4.96 | 3 | 1257 | 5.2785e+0 | 2.6042e+1 | 4.93 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |||
| 9 | 20447 | 7.4173e−1 | 3.6179e+0 | 4.88 | 9 | 19876 | 7.4923e−1 | 3.6707e+0 | 4.90 |
| 10 | 36448 | 5.3738e−1 | 2.6609e+0 | 4.95 | 10 | 35236 | 5.4581e−1 | 2.7078e+0 | 4.96 |
The history of errors
Convergence plots of indicators
To examine quadratic convergence behavior in the sense of (2.5) for
The ratios
The history of quadratic convergence and various differences for Example 6.1
Meshes and
Graph of
We present the adaptive meshes
Example 6.2 (A Solution with a Corner Singularity)
We consider SEE (4.1) with
specifies data 𝑓.
On the Dirichlet boundary
We note that
To take the Neumann part of the boundary condition into account, we modify indicator
Convergence orders (CO) of various errors on uniform meshes for Example 6.2
| Multi-level algorithm | Nonlinear solver | ||||||||
|---|---|---|---|---|---|---|---|---|---|
|
|
|||||||||
| 𝑗 |
|
|
CO |
|
CO |
|
CO |
|
CO |
| 0 | 96 | 2.3413e−2 | 2.3451e−1 | 2.3413e−2 | 2.3451e−1 | ||||
| 1 | 384 | 9.7023e−3 | 1.27 | 1.4761e−1 | 0.67 | 9.7329e−3 | 1.27 | 1.4760e−1 | 0.67 |
| 2 | 1536 | 3.9676e−3 | 1.29 | 9.1644e−2 | 0.69 | 3.9724e−3 | 1.29 | 9.1644e−2 | 0.69 |
| 3 | 6144 | 1.5829e−3 | 1.33 | 5.6561e−2 | 0.70 | 1.5836e−3 | 1.33 | 5.6561e−2 | 0.70 |
| 4 | 24576 | 6.2119e−4 | 1.35 | 3.4866e−2 | 0.70 | 6.2129e−4 | 1.35 | 3.4866e−2 | 0.70 |
| 5 | 98304 | 2.4160e−4 | 1.36 | 2.1517e−2 | 0.70 | 2.4161e−4 | 1.36 | 2.1517e−2 | 0.70 |
Convergence orders of errors and etas on adaptive meshes for Example 6.2
Errors and etas, and efficiency indices on adaptive meshes for Example 6.2
| 𝑗 |
|
|
|
|
|---|---|---|---|---|
| 0 | 96 | 2.3451e−1 | ||
| 1 | 384 | 1.4761e−1 | 4.8126e−1 | 3.26 |
| 2 | 415 | 1.1532e−1 | 3.7244e−1 | 3.23 |
| 3 | 548 | 8.9599e−2 | 2.8853e−1 | 3.22 |
| 4 | 991 | 6.5174e−2 | 2.0990e−1 | 3.22 |
| 5 | 1875 | 4.6944e−2 | 1.5114e−1 | 3.22 |
| 6 | 3705 | 3.3205e−2 | 1.0794e−1 | 3.25 |
| 7 | 7177 | 2.3774e−2 | 7.7373e−2 | 3.25 |
| 8 | 13858 | 1.7006e−2 | 5.5668e−2 | 3.27 |
The convergence of indicators
Uniform convergence
Adaptive convergence
Uniform ratios
Adaptive ratios
Meshes and
Graph of
We only provide results for
In Figure 7 (a) and (b), we present indicators and their convergence orders on uniform and adaptive meshes, respectively.
The convergence order of additional term
We present adaptive meshes
6.2 Navier–Stokes Equations
Now, we consider Navier–Stokes equations (3.1)–(3.3).
We use the lowest-order Raviart–Thomas finite element space
We define the local error estimator
the global error estimator
and efficiency indices
Example 6.3 (A Solution with a Boundary Layer)
We consider NSE (3.1)–(3.3) with
specify data 𝒇 and 𝒈.
Table 5 shows convergence history of multi-level algorithm and nonlinear solver on uniform meshes.
We observe that the errors
Convergence orders (CO) of errors on uniform meshes for Example 6.3
| Multi-level algorithm | Nonlinear solver | ||||
|---|---|---|---|---|---|
|
|
|||||
| 𝑗 |
|
|
CO |
|
CO |
| 0 | 256(8) | 7.903738e+0 | 7.903738e+0 | ||
| 1 | 1024(16) | 4.556065e+0 | 0.795 | 4.556080e+0 | 0.795 |
| 2 | 4096(32) | 2.436524e+0 | 0.903 | 2.436521e+0 | 0.903 |
| 3 | 16384(64) | 1.245096e+0 | 0.969 | 1.245096e+0 | 0.969 |
| 4 | 65536(128) | 6.262103e−1 | 0.992 | 6.262102e−1 | 0.992 |
| 5 | 262144(256) | 3.135742e−1 | 0.998 | 3.135742e−1 | 0.998 |
Efficiency indices on adaptive meshes for Example 6.3
| 𝑗 |
|
|
|
|
|---|---|---|---|---|
| 0 | 256 | 7.903738e+0 | ||
| 1 | 1024 | 4.556065e+0 | 1.164e+1 | 2.554 |
| 2 | 1206 | 2.472931e+0 | 4.728e+0 | 1.912 |
| 3 | 1486 | 1.525288e+0 | 2.924e+0 | 1.917 |
| 4 | 2260 | 1.137552e+0 | 2.025e+0 | 1.780 |
| 5 | 3492 | 7.797460e−1 | 1.455e+0 | 1.866 |
| 6 | 6130 | 5.987671e−1 | 1.066e+0 | 1.780 |
| 7 | 10614 | 4.118791e−1 | 7.717e−1 | 1.874 |
| 8 | 20130 | 3.183675e−1 | 5.678e−1 | 1.783 |
| 9 | 35736 | 2.221120e−1 | 4.165e−1 | 1.875 |
Various behaviors for Example 6.3
Convergence of indicators on uniform mesh
Convergence of indicators on adaptive mesh
Quadratic convergence behavior
Meshes and
Graph of
We present results for the adaptive multi-level algorithm applied to Example 6.3 with a boundary layer in Table 6.
The superiority of the adaptive multi-level algorithm can be clearly seen in Table 5 and Table 6.
Indeed, the accuracy on uniform mesh with
Figure 9 (b) and (c) illustrate the history of indicators on uniform and adaptive meshes, respectively.
Here, the indicators
We observe that the additional term
To examine the behavior of quadratic convergence in the sense of (2.5), we investigate the difference between
We find that the ratios
In Figure 10, we exhibit some meshes
7 Conclusion
The adaptive multi-level algorithm analyzed in this paper can be used to efficiently solve nonlinear problems with singularity or layer. We start with a nonlinear system on a very coarse mesh and then solve linearized system on the successively refined meshes with adaptivity. The BRR framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the nonlinear problem.
As applications, the Navier–Stokes equations and the semilinear elliptic equations are considered. Residual type reliable and efficient a posteriori error estimators are constructed for the adaptive multi-level algorithm for the considered problems. With few modifications we apply existing a posteriori error estimators for the linear Stokes equations and elliptic equations corresponding to the NSE and SEE, respectively. Note that the multi-level algorithm with uniform meshes has quadratic convergence in the sense of (2.5). We emphasize that the multi-level algorithm with adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is validated from the numerical results presented in this paper.
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: NRF-2021R1F1A106243413
Award Identifier / Grant number: NRF-2022R1A2B5B02002481
Award Identifier / Grant number: NRF-2020R1I1A1A01070361
Funding statement: Dongho Kim was supported by NRF-2021R1F1A106243413. The research of Eun-Jae Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (NRF-2022R1A2B5B02002481). Boyoon Seo was supported by NRF-2020R1I1A1A01070361.
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