A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations ()
1. Introduction
Mixed finite element method is a kind of method in solving partial differential equations (PDEs). Mixed methods are based on writing a higher order differential equation into lower order differential system. The purpose of this article is to study the a posteriori error estimate of the mixed finite element methods for the following semilinear elliptic equation
(1.1)
with mixed boundary conditions.
is a convex polygonal domain and
, a real-valued function on
, has continuous first and second derivatives to u.
The a posteriori error estimates of mixed finite element method have been studied extensively in the past several decades for solving many differential model problems, for example, the Navier-Stokes equations based on Newton-type linearization by Durango and Novo [1], the linear elliptic problems by Larson and Målqvit [2], the Poisson problem about an error estimate in the
norm of the flux by Carstensen [3], the general convex optimal control problems by Chen and Liu [4]. In order to combine the advantage of adaptive mixed finite element method and the efficiency of two-grid finite element method for semilinear elliptic equations, in this study, we proposed the posteriori error estimator for the two-grid mixed finite element methods.
The two grid method is a widely used numerical method in solving nonlinear problems. It was first introduced by Xu [5] [6] to solve the nonsymmetric linear and nonlinear elliptic problems. Many numerical methods combined with two-grid method were used to solve different model problems, for instance, nonlinear reaction-diffusion equations using mixed finite element methods by Chen and Chen [7], nonlinear parabolic equations by Chen and Liu [8], the coupled Stokes-Darcy system by Sun, Shi, et al. [9], two-dimensional semi-linear elliptic interface problems by Chen, Li, et al. [10]. In recent years, the residual-based a posteriori error estimates of two-grid finite element methods and finite volume methods are investigated for nonlinear PDEs [11] [12]. Adaptive two-grid finite element methods based on residual-based a posteriori error estimator are studied in [13].
In order to investigate efficient two-grid adaptive mixed finite element method for semilinear or nonlinear PDEs, in this paper, we study two-grid mixed finite element method and its posteriori error estimates for semilinear elliptic problem (1.1). We first propose two algorithms for the model problem. Then, for both two-grid mixed finite element methods, by using averaging technique, the posteriori error estimators are proposed for the flux error in L2-norm. Theoretical analysis is given to prove the efficiency and reliability of the error estimators. Two numerical examples are given to verify the theoretical results and from the numerical results, we find that the error estimators proposed in this paper are efficient and reliable.
The outline of this paper is organized as follows. In Section 2, we present some notations and weak form of the semilinear elliptic Equations (1.1). Two-grid mixed finite element methods for the model problem are presented in Section 3. In Section 4, we give a theoretical analysis of the reliability and efficiency for the posteriori error estimators. Numerical experiments are given to verify the theoretical results in Section 5.
2. Weak Form and Preliminaries
In this section, we will present some preliminaries and weak form for the semilinear model problem (1.1).
2.1. Preliminary
We first introduce the standard notations used in this paper. We denote
as Sobolev spaces with the norm
, for integer
and real number
,
and
. When
, we denote
by
,
by
, and we will use
and
.
Throughout this paper, we will use letter C to denote a generic positive constant that may represent different values at different places.
2.2. Weak Form
In order to introduce a mixed variational formulation on
, we first introduce the following spaces
with the norms
The Lipschitz boundary
of the bounded domain
is split into a closed Dirichlet part
and possibly empty Neumann part
. Set
. Rewrite the problem (1.1), we have
(2.1)
Here
is given function.
We define a space
, the standard mixed variational form of (2.1) is to find
, such that
(2.2)
(2.3)
where
is the inner product of
.
Given
, the linearised form of (2.2) and (2.3) is
(2.4)
(2.5)
Let
denote a regular triangulation of the polygonal domain
,
denotes the diameter of the element
and
. And for
, let
denote the set of algebraic polynomials in
variables on T of total degree ≤ k.
The set of all nodes and edges appearing in
are denoted as
and
.
denotes edges
on the boundary
,
denotes edges
on the boundary
,
denotes edges
but
.
The space
(possibly discontinuous) of
-piecewise polynomials of degree ≤ k is the set of all
(a set composed of all bounded number columns) with
for all T in
. Set
Here
denotes continuous space.
Let
In this paper, we mainly study
as the lowest order Raviart-Thomas mixed finite element spaces for the discretization of the flux
and u, we define
(
has the same definition as
). Therefore, the discretization of mixed finite element method is to find
such that
(2.6)
(2.7)
2.3. Helmholtz Decomposition and Interpolation Operator
In order to make theoretical analysis, we need to introduce Helmholtz decomposition and the interpolation operator
[14].
We first define the curl operator as follows [15],
Then, we can get
and the Gauss theorem yields the following relation
(2.8)
here
is outer unit normal vector of
, and
denotes the tangential component of
.
We define an approximation operator
. Let
denote the nodal basis of
,
satisfies
if
and
, the open patches defined by
.
Then, we modify
to be a partition of unity
(
denotes the set of free nodes). Find each fixed node
, we choose a node
and let
if
. In this way, we define a partition of
into
classes
, where
. For each
, set
and notice that
is a partition of unity. It is required that
is connected.
For
and
, let
be
and then define
We also define local mesh-sizes by
and
, where
denotes the element-size,
for
, and the edge-size
.
We also use the orthogonal
-projection
[16] :
, which satisfies
(2.9)
Lemma 2.1 ( [14] ). There exist
-independent constant C such that for all
and
, there holds
(2.10)
(2.11)
(2.12)
(2.13)
the constants C only depend on
,
,
and the shape of the elements and patches.
3. Two-Grid Finite Element Methods for Semilinear Problems
In this section, we present two-grid mixed finite element methods for the semilinear elliptic problems and analyze the lower and upper bounds of posteriori error estimates by averaging techniques.
The idea of two-grid methods is to solve the semilinear partial differential equations(PDEs) on the coarse mixed finite element spaces
first and then find the solution (
) (or (
)) of a linear PDEs on the finer mixed finite element spaces
. The basic mechanism in these algorithms is to construct two shape-regular subdivision of
as
and
with different mesh sizes H and h (
).
Two-grid Algorithm 1
Step 1: On the coarse mesh
, compute
to satisfy the following original nonlinear system:
(3.1)
(3.2)
Step 2: On the fine grid
, compute
to satisfy the following linear system:
(3.3)
(3.4)
The second two-grid algorithm introduces the Newton linearized procedure on the fine mesh to linearize the semilinear system.
Two-grid Algorithm 2
Step 1: On the coarse grid
, compute
to satisfy the following original nonlinear system:
(3.5)
(3.6)
Step 2: On the fine grid
, compute
to satisfy the following linear system:
(3.7)
(3.8)
For semilinear system (3.1) and (3.2) and (3.5) and (3.6), we use the Newton iteration to compute
in the implementation.
In averaging techniques, the error estimator is based on a smoother approximation in
, the continuous
-piecewise linears approximation to the discrete solution
(or
), for instance,
which can be served as a computable estimator.
The triangle inequality shows that
is efficient up to higher order terms of exact solution
, indeed,
The last term converges as
is of higher order than the error
. So we have
(3.9)
where “h.o.t.” denotes the higher-order term.
In the following, by using the solution (
) (or (
)), Helmholtz decomposition and interpolation operator
, we analyze the upper bound of
and
for the two algorithms.
3.1. A Upper Bound for the Error of Two-Grid Algorithm 1
By Helmholtz decomposition, we get the following lemma.
Lemma 3.1 ( [14] ). There exist
that satisfy boundary condition
and
is constant
(3.10)
and then
(3.11)
In order to estimate the right hand side of (3.11), and using the theoretical analysis in [17], we have
Lemma 3.2. Suppose the u and
are the solutions of (2.2) and (2.3) and (3.1) and (3.2), there exists a constant C independent of H such that
(3.12)
We can also get
.
Then, by using the Green’s fromula and Lemma 3.2, we can bound the first contribution of (3.11).
Lemma 3.3. Let
and
are the solutions of (2.2) and (2.3) and (3.3) and (3.4), and
. Then we have
(3.13)
Proof. Employ the Green’s formula and
-projection
, we can get the first contribution on the right-hand side of (3.11), that
(3.14)
Here mean value
of
. Now we estimate the right-hand side terms. For
, using Lemma 3.2, we conclude that
here
is a value between u and
. For
, employ the inequality (2.9), and
with Lemma 4.2 of [14], we conclude that
(3.15)
which completes proof.
Lemma 3.4. Suppose
,
, and
is the two-grid solution satisfying (3.3) and (3.4), for
and
, there holds
(3.16)
Proof. Employ the Green’s formula, then,
(3.17)
Employ the inequalities (2.10), (2.11) and (2.13) of Lemma 2.1, the first two terms of the right hand side of (3.17) can be written
(3.18)
(3.19)
(3.20)
And here we have
, so we estimate the last term
just like estimate
, and by using triangle inequality. Therefore, we complete the proof.
For the second term on the right hand side (3.11), denote
for vectors, then,
and use that
, let
and
, use
for all
, where
. Therefore, we can get the following lemma.
Lemma 3.5. Let
and
are the solutions of (2.2) and (2.3) and (3.3) and (3.4), then,
(3.21)
Proof. Using Lemma 3.2 and Lemma 3.4 and (2.8), let
on
. So we have
(3.22)
By using Lemma 2.1, we get
(3.23)
(3.24)
where
denotes unit tangential vector, using
, (3.23) can be written as
(3.25)
the last term
similar to
of Lemma 3.4, which completes the proof.
Combine with Lemma 3.3 and Lemma 3.5, and using Lemma 5.2 of [14], we can know
and
vanished, we have the following conclusion
Lemma 3.6. Let
and
are the solutions of (2.2) and (2.3) and (3.3) and (3.4),
, then, we have,
(3.26)
The last three terms are high order terms compared with error
. By using the inverse inequality, we have the following upper bound result
Lemma 3.7. Suppose that the discrete
satisfies
,
, then
(3.27)
Combined with the efficiency (3.9) of the estimator, we have
Theorem 3.1. Let
be the exact solution of (2.2) and (2.3) and
be the solution of (3.3) and (3.4), then, we have
(3.28)
3.2. A Upper Bound for the Error of Two-Grid Algorithm 2
In this subsection, we will get the upper bounds for Algorithm 2. In order to make a theoretical analysis of Algorithm 2, we need to assume that the first derivative of
satisfies
in this subsection. First, we need the following priori error estimate for approximate solution
from Algorithm 2.
Lemma 3.8. Let u be the exact solution of (2.2) and (2.3) and
be the solution of (3.7) and (3.8), then, we have,
(3.29)
Proof. Using (2.2) and (2.3) and (3.7) and (3.8), we have
Using Taylor expansion for
at
, let
,
and add the last two equations together to get
(3.30)
where
is some value between u and
. By using the assumption of
as well as the Cauchy inequality, we have the following estimation
which completes the proof.
Lemma 3.9. Let
and
are the solutions of (2.2) and (2.3) and (3.7) and (3.8). Then
(3.31)
Proof. Employ the Cauchy’s inequality and Poincaré inequality, we have
(3.32)
here
is the average of
. Similar to
, we can estimate
as
Here
is some value between u and
, which completes the proof.
Similar to Lemma 3.4, we have the following result.
Lemma 3.10. Let
and
are solutions of (2.2) and (2.3) and (3.7) and (3.8), then, we have
(3.33)
Lemma 3.11. Let
and
are the solutions of (2.2) and (2.3) and (3.7) and (3.8), then, we have
(3.34)
Combine with Lemma 3.9 and Lemma 3.11, and similar to Lemma 3.6, we finally get the following result.
Lemma 3.12. Let
and
are the exact and numerical solutions satisfying (2.2) and (2.3) and (3.7) and (3.8) respectively, then,
(3.35)
By Lemma 3.12 and similar efficiency result (3.9), we have.
Theorem 3.2. Suppose
satisfies
and
, then,
(3.36)
4. Posteriori Error Estimator with Averaging Technique
In this section, we use the averaging technique to construct an averaging operator
of posteriori error estimator, and prove
(or
) is very approximation to
(or
). In practise, we use the averaging operator
to compute the upper bound of the
(or
). Take Algorithm 1 for instance (all the following conclusions are also hold for Algorithm 2). Set
then, the minimum
is frequently replaced by an upper bound
,
From [18], one of a popular averaging operator
is defined, for each node z, by
where
with
being an orthogonal projection and linear and continuous averaging
being defined as
where
denotes the area of patch
related to node z.
For the efficiency of estimator
, we have the following result.
Lemma 4.1 ( [18] ). There exists a mesh-size independent positive constant C with
According to the relationship between
and
, we conclude the following result.
Theorem 4.1. Let
and
are the exact solution and numerical solutions from Algorithm 1 respectively, then, we have
(4.1)
The result also holds for
, i.e.,
(4.2)
5. Numerical Experiments
In this section, we will validate the a posteriori error estimates of averaging technique of two-grid mixed finite element solutions for semilinear elliptic equations by some numerical examples. Our focus is to deserve the ability of the error estimates to imitate the convergence behaviour of the error in the
-norm.
To simplify our problem, we consider the following semilinear elliptic equations with entire boundary
:
(5.1)
And use the lowest order Raviart-Thomas mixed finite elements in the implementation. We consider the mesh
satisfy
in the following numerical experiments.
Example 1: We choose
in a way such that the exact solution:
so we get the explicit expression of
as:
with the domain
.
Example 2: We choose
in a way such that the exact solution:
so we get the explicit expression of
as:
with the L-shape domain
.
From the numerical results presented in Table 1 and Table 2 for Example 1 and Table 3 and Table 4 for Example 2, we conclude that error estimators by averaging technique is reliable and efficient for both Algorithm 1 and Algorithm 2.
Table 1. A posteriori error of the Algorithm 1 for the example 1.
Table 2. A posteriori error of the Algorithm 2 for the example 1.
Table 3. A posteriori error of the Algorithm 1 for the example 2.
Table 4. A posteriori error of the Algorithm 2 for the example 2.
6. Conclusion
In this paper, we present a posteriori error estimate of two-grid mixed finite element method for semilinear elliptic equations by using averaging techniques. Theoretical analysis as well as the numerical experiments is provided to prove the efficiency and reliability of error estimators. In our following work, we will construct the adaptive two-grid mixed finite element method for the semilinear elliptic equations using the posteriori error estimators studied in this paper.