Abstract
The Uzawa method is an iterative approach to find approximated solutions to the Stokes equations. This method solves velocity variables involving augmented Lagrangian operator and then updates pressure variable by Richardson update. In this paper, we construct a new version of the Uzawa method to find optimal numerical solutions of the Stokes equations including corner singularities. The proposed method is based on the dual singular function method which was developed for elliptic boundary value problems. We estimate the solvability of the proposed formulation and special orthogonality form for two singular functions. Numerical convergence tests are presented to verify our assertion.
1. Introduction
To study the solution of a partial differential equation, the equation is sometimes interpreted in a weak (variational) sense and we can define the regular problem in this manner. Consider a variational problem:where and is bilinear form with continuity and coercivity. We call the problem is -regular if, for every , there is a solution with a constant such that
It is well known that the approximated solution of regular problems shows optimal convergence by using common numerical methods such as the finite difference method or the finite element method.
However, if a computational domain of an elliptic boundary value problem is a polygon including reentrant corners, then the problem is not -regular, , and it is sometimes called a corner singular problem, and the singular solution is hard to approximate optimally. Therefore, there have been many numerical approaches developed to solve the problem efficiently. These methods aim to improve accuracy and to resolve the convergence difficulties. The version of the finite element method is a typical method. It employs elements of variable sizes and polynomial degrees to improve the convergence rate and accuracy in [1].
The postprocessing method has been proposed in [2–6]. These methods calculate the coefficients of the singular coefficients from the finite element solution. Singular function boundary integral method also has been proposed to treat singular problems in [7–9]. These methods calculate the unknown coefficients of singular functions directly. The solution is approximated by the leadinig terms of the local asymptotic solution expansion, and the Dirichlet boundary conditions are weakly enforced utilizing Lagrange multipliers.
It is well known that the solution of an elliptic boundary value problem can be decomposed of a finite number of so-called singular functions, which come from the neighborhood of a corner point, and a smooth remainder function which is called a regular solution (see, for instance, [10–12]). The singular function method is one approach to use this fact. It takes into account the form of the singular solution by finding stress intensity factors and regular solution (see, for example, [13–15]). These methods consist of augmenting the spline space by the singular functions. The dual singular function is based on the observation that the regular part of the solution and the stress intensity factor are related to each other by testing a dual singular function in [16–18]. It was implemented as an iterative procedure that iterates back and forth between these equations. Also, this approach was extended to multigrid versions.
In [19], Cai and Kim developed and analyzed a finite element method for the accurate computation of the solution and intensity factors. If a regular part of the solution is smoother than the solution itself, then approximated solutions of a standard finite element lose accuracy. Therefore, the new method finds approximated regular solution first and then computes stress intensity factors and solution. This method decoupled variational formulations by testing the dual singular function and disjoint cut-off function. Consequently, the regular solution is uniquely determined by a well-posed variational problem, and stress intensity factors can be expressed by a regular solution. We call this method the finite element dual singular function method (FE-DSFM).
This method was extended to other problems. Poisson problems with mixed boundary conditions and interface problems were applied in [20, 21]. Also, this algorithm was studied to corner singularities of the Helmholtz equation and heat equation (see [22, 23]). Some studies have been designed to target the singular solution of the Stokes equation which is much more complicated. We refer the interested reader to the papers [24, 25] for a formula for corner singularity of the Stokes equations. A mixed finite element method-based FE-DSFM was developed for Stokes equations in [26, 27]. These proved the accuracy and well-posedness of the algorithm including two singularities at each corner.
In this paper, we proposed a new algorithm for solving Stokes equations. The proposed algorithm is based on the Uzawa algorithm and finite element method in [28]. We estimate the solvability of the proposed formulation and special orthogonality form for two singular functions. We also give numerical convergence tests to verify our assertion.
2. Singular Functions of Stokes Equations
Let the computational domain be an open and bounded concave polygon in especially having one reentrant corner. The steady-state Stokes equation iswhere is a given external force field causing an acceleration of the flow in , is a computational domain in , and is the reciprocal of the Reynolds number. The unknowns are the (vector) velocity field and the (scalar) pressure . The pressure gradient plays a role in an additional force, which prevents a change in the density. In particular, high pressure builds up at points, where, otherwise, a source of the sink would be created. Mathematically, the pressure can be considered as a Lagrange multiplier. Besides, the weak formulation of the Stokes equations leads to a saddle point problem with the restriction .
There are mainly two approaches to find finite element approximation of solutions of (3). One approach is called the mixed finite element method which solves velocities and pressure simultaneously by constructing a big linear system. Another approach is an iteration method called the Uzawa method which solves velocity variables involving augmented Lagrangian operator and then updates pressure variable by Richardson update. The advantage of the Uzawa method is that it uses less memory because it solves velocity and pressure separately. However, the iteration process sometimes takes more computational time. The Uzawa iteration can be extended to projection-type methods such as the gauge-Uzawa method to solve unsteady incompressible Navier–Stokes equations.
If the solution of (3) is smooth enough, namely, with , and if a suitable finite element pair is imposed for velocity and pressure, then the finite element solution using the standard mixed finite element method has optimal error bounds as shown in [14]:where is the biggest mesh size. However, if , then the error bounds only become
For the case , we call the solution a singular solution, otherwise a regular solution. Since singularities are due to re-entrant corners of a computational domain , we assume that is open and bounded polygonal domain in with one re-entrant corner.
To derive singular and dual singular functions for the Stokes equations, the polar coordinate of homogeneous Stokes system (3) should be considered, and we can find the singular and dual singular solution via solving the homogeneous system with the separation of variables. Then, we arrive at the singular function of system (3):where is solution ofand and are
Similarly, the dual singular functions are derived bywhere is solution of (7) and and are
The following lemma describes the number of singular functions depending on the value of and the relationship between values of and is shown in Figure 1.
Lemma 1. (see [26]). For , we have the following properties:(1) and are solutions for any . We call these trivial solutions.(2)There are only trivial solutions for . It means that there is no , solution of Stokes equation with homogeneous boundary condition on , where is the boundary near re-entrant corner.(3)If , then there is a unique solution except trivial solutions. Therefore, there is one singular solution .(4)If , then there are two solutions except trivial solutions. Thus, there is two singular solutions .(5)If , then solution is except trivial solutions.
It is shown that 2 singular functions of solution (3) can be involved in each reentrant corner. If there is one re-entrant corner, the solution of (3) can be written by the formwhere and are the stress intensity factors, , , are singular functions of (6), and is the regular solution.
Lemma 2 (see [26]). The singular function and the dual singular function , , satisfyrespectively. The boundary conditions of and vanish on , but the boundary value of is not defined at the origin. Both of and are not on .
3. FE-DSFM and Uzawa Iteration
Uzawa method solves velocity variables involving augmented Lagrangian operator and then updates the pressure variable by Richardson update. Algorithm 1 is the Uzawa method for solving stokes equation (3).
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There are some theoretical works of literature on the Uzawa method and augmented Lagrangian method. In [29–32], the convergence of Algorithm 1 is proved. Especially, it is shown that and have optimal convergence in [31]. So, we choose and for our algorithms.
Our goal is to construct FE-DSFM with the Uzawa iteration. We consider Stokes equations with polygonal domain including one reentrant corner. To represent FE-DSFM, we first introduce the cut-off function. Let be the internal angle of satisfying . Now, we setand in the polar coordinates and define cut-off functionwhere is very smooth function, is a parameter in , and is a fixed real number which will be determined later so that has 0 on whole .
We note that the singular function (6) () are the solutions of homogeneous Stokes equation (3) with vanishing Dirichlet boundary condition on . Also, has to be a positive real number and , . Let be a smooth cut-off function (14) with smooth function which is equal to the one identically in the neighborhood of origin, and the support of is small enough so that the functions , , vanish identically on . Then, in general, the solution including singular parts of (3) can be rewritten in the form (11):where and are the stress intensity factors and .
The strategy of FE-DSFM is to find the regular solution and stress intensity factors and by applying the standard Uzawa method with an additional equation. To make simpler formula, we assume that there is only one singular solution, that is, one of or is zero. Let and . Then, Step 1 of the Uzawa algorithm can be
Even though and , we know that . So, (16) can be rewritten as
For the sake of a clear explanation, we note that the inner product of vectors and is
Then, the weak formulation of (17) isfor all . Since and , equation (19) is solvable, and it has unique solution if we know . However, since is unknown, we need one more equation, and it should be(i)A single equation because is a real number(ii)Linearly independent to (19)(iii) which is not disappeared(iv)Solvable near the singularity corner
Since equation (19) is a Poisson-type problem to find , we can make the additional equation by testing the dual singular function of the following Poisson equation, instead of that of Stokes equation (9):
Lemma 3. (see [19]). The function in (20) has the following properties:(1) is not defined at origin(2) and (3)(4) (except origin)(5)For all , We denote the vector form of (20), . We now test to (17) and apply Lemma 3 to obtainLet and . Then, we have the system
We first check solvability of system (22).
Lemma 4. Define a mapping and such that , whereGiven a function ,
Therefore, there is a unique value such that for all .
Proof. Let with , , and . Define two equations:Then, and . Then,We note that in and so is with on . Since in , we have
Theorem 1. Formulation (22) has a unique solution in .
Proof. By Lemma 4, contract mapping theorem provides the existence of the unique fixed point of .
We now define finite element space similarly as in previous sections to construct fully discrete FE-DSFM. Let be a shape-regular quasi-uniform partition of of mesh size into closed elements . The vector and scalar finite element spaces arewhere and are spaces of polynomials with degree bounded uniformly with respect to . We stress that the space is composed of continuous functions to use integration by parts:for all . Then, the finite element approximation for (22) is to find and such thatwhere . And, the matrix form of the coupled system (30) becomeswhere is symmetric positive definite square matrix and are column vectors. System (31) can be solved by the Sherman–Morrison formula:and can be computed by second equation of (30).
Sherman–Morrison formula is a rigorous approach to solve system (30), but it is sometimes complicated to apply for some problems including many singular functions. Since by Lemma 4, we choose instead of in first equation of (30). Algorithm 2 is the proposed method for one singular function.
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Remark 1. Compared with the original Uzawa iteration (Algorithm 1), the proposed method (Algorithm 2) needs only one more linear solver for the initial process.
4. Algorithm for Two Singular Functions
In this section, we construct an algorithm for two singularities in one corner. We consider the solution is
With the implicit formation of two stress intensity factors, step 1 of Uzawa algorithm is
Here, the unknowns are , , and . However, our test function which is not in is only . Instead of finding another test function, we use the following property.
Theorem 2. (properties of singular functions). Let and be the singular functions of Stokes equation on same corner. Then,
Proof. To tell the conclusion, it is too difficult to show equation (35) by algebraic calculation because the integral functions depend on some constants which are constructed by solutions of transcendental function.
First of all, from singular function (6) and cut-off function (14) with smooth function, we obtainLetThen,SinceWe can conclude that equation (35) holds ifHowever, we could not show equations (40)–(43) are valid by algebraically computation. Therefore, we calculate them numerically.
For numerical substitution, we first compute the above integral forms:We remark that the values and are from the equationand it is not possible to solve directly. So, we use the numerical root-finding method and bisection method for various tolerances of approximated . For internal re-entrant corner angle , we choose 400 angles from to by interval. Denote that(i) left side of equation (40)(ii) left side of equation (41)(iii) left side of equation (42)(iv) left side of equation (43)We choose three tolerances , , and for solving and . By substituting , , and to (44)–(47). Figure 2 shows numerical substitution values of , , , and . We can see that the values strongly depend on the tolerances. If and are exact real values, we can claim our assertion.
Now, we construct the Uzawa method for two singular solutions. We assume that the form (33) is valid. The weak formulation of Step 1 in the Uzawa method test by is in the formLet and . Then, component of (49) isand component of (49) isIf we use the following notationsthen (50) and (51) can be rewritten byTherefore, we obtainFinally, combining (54) and (55) and equation (35) in Theorem 2 yieldsAs a result, we can calculate one stress intensity factor bySimilarly, another stress intensity factor can be computed by
(a)
(b)
(c)
(d)
Remark 2. Since , Lemma 4 is valid for two singular functions case.
Algorithm 3 is the proposed Uzawa iteration for two singularities.
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