## 1 Introduction

Schemes of domain decomposition are considered for numerical solving time-dependent problems to partial differential equations. Iteration-free algorithms of domain decomposition take into account specific features of time-dependent problems in the most efficient way. In some cases, it is possible (see, e.g.,

Kuznetsov (1988, 1991)) without loss of accuracy of the approximate solution make only one iteration of the Schwarz alternating method at a new time level in solving boundary value problems for a parabolic second-order equation. Iteration-free schemes of domain decomposition are associated with various variants of additive schemes (splitting schemes) – see regionally additive schemes in Samarskii et al. (2002).Methods of domain decomposition for solving unsteady problems can be classified by (I) the method of decomposition for a calculation domain, (ii) the choice of decomposition operators (exchange boundary conditions), and (III) the splitting scheme (approximation in time) employed. For multidimensional boundary value problems, it is possible to use domain decomposition methods with or without overlapping subdomains

(Quarteroni and Valli, 1999; Toselli and Widlund, 2005). Methods without overlapping subdomains are connected with an explicit formulation of exchange data conditions on subdomain interfaces.To construct decomposition operators for solving unsteady initial boundary value (IBV) problems for partial differential equations, it is convenient to use a partition of unity for a computational domain (Dryja, 1991; Laevsky, 1987; Samarskii and Vabishchevich, 1995; Vabishchevich, 1989, 1994a, 1994b). In decomposition methods with overlapping subdomains, the functions are associated with individual subdomains and take a value between zero and one. Results of studies on domain decomposition methods for Cauchy problems for partial differential equations are summarized in the book Samarskii et al. (2002). Among more recent studies, we highlight the works Vabishchevich (2008, 2011), where schemes of domain decomposition more suitable for numerical implementation are presented.

The construction of regionally additive schemes and study of their convergence are carried out on the basis of the general theory of splitting schemes (Samarskii, 2001; Marchuk, 1990; Vabishchevich, 2014). We highlight the simplest case of two-component splitting. In this case, we obtain unconditionally stable factorized splitting schemes, such as classical methods of alternating directions, predictor-corrector schemes and so on. A more interesting for computational practice is the situation, when a problem operator is divided into a sum of three or more noncommutative non-self-adjoint operators. In the case of such a multicomponent representation, splitting schemes are constructed on the basis of the concept of summarized approximation. For parallel computers, additively averaged splitting schemes are of particular interest. In the class of splitting schemes with full approximation (Vabishchevich, 2014)

, we highlight vector additive schemes, when the original equation is transformed into a system of similar equations

(Abrashin, 1990; Vabishchevich, 1996; Abrashin and Vabishchevich, 1998). The most convenient approach for constructing additive operator–difference schemes of multicomponent splitting is based on regularization of difference schemes (Samarskii, 1967), where stability is achieved via perturbations of operators of the difference scheme.The regionally additive schemes constructed on the basis of a partition of unity have certain defects. The most important among them is connected with the use of the uniquely defined operators with the generating coefficients in the overlapping domains. This leads to the fact that, for example, in the problems with constant coefficients of the equations one needs to use an algorithm with varying coefficients. In the paper Vabishchevich (2008), we constructed unconditionally stable regionally additive schemes with overlapping subdomains, which are more comfortable for the practical use than traditional schemes designed on the basis of a partition of unity. The regularization principle for operator–difference schemes (Samarskii, 1967) makes possible to construct schemes of component-wise splitting. In the present work, domain decomposition methods with overlapping subdomains are designed using a two-component splitting, which generalize standard factorized schemes (Samarskii et al., 2002; Vabishchevich, 2014). Decomposition operators are constructed on the basis of the indicator functions of the subdomains.

The paper is organized as follows. A model problem for a parabolic equation with self-adjoint elliptic operator of second order is formulated in Section 2. Approximation in space is constructed using Lagrangian finite elements, whereas approximation in time is based on conventional two-level schemes with weights. In Section 3, a two-component scheme of domain decomposition with overlapping subdomains is constructed on the basis of a partition of unity for the computational domain. New schemes of domain decomposition developed using indicator functions of subdomains are proposed in Section 4. They are based on the generalization of classical factorized schemes (operator analogs of ADI schemes). In Section 5, numerical experiments on the accuracy of the domain decomposition schemes are discussed for the model IBV problem. The results of the work are summarized in Section 6.

## 2 Problem formulation

Let be a bounded domain () with a piecewise smooth boundary . We define an elliptic operator so that

(1) |

on the set of functions

(2) |

Let and be smooth functions in and

The Cauchy problem

(3) |

(4) |

is considered with , using the notation .

Let be the scalar product and norm in , respectively:

A symmetric positive definite bilinear form such that

is associated with the Hilbert space , where the scalar product and norm are, respectively:

Define as a subspace of such that

Multiplying (3) by and integrating over the domain , we arrive at the equality

(5) |

Here is the following bilinear form:

where

In view of (4), we put

(6) |

The variational (weak) formulation of the problem (1)–(4) consists in finding , that satisfies (5), (6) with the condition (2) on the boundary.

For the solution of the problem (5), (6

), we have the a priori estimate

(7) |

To show this, we put in (5) and get

Taking into account the positive definiteness of the form and the inequality

we obtain

From this inequality, in view of (6), it follows that the estimate (7) holds.

It is easy to obtain less trivial and more interesting a priori estimates for the solution of the problem (5), (6). We confine ourselves to the elementary estimate (7) aiming at obtaining similar estimates using various approximations in time and in space for the problem under consideration.

For numerical solving the IBV problem (3), (4), approximation in space is constructed using the finite element method. The weak formulation (5), (6) is employed. Define the subspace of finite elements and the discrete elliptic operator as

The operator acts on the finite dimensional space and

(8) |

where is the identity operator.

For the problem (3), (4), we put into the correspondence the operator equation for :

(9) |

(10) |

where , with denoting -projection onto .

Let us multiply equation (9) by and integrate it over the domain :

In view of the self-adjointness and positive definiteness of the operator , we have

The right-hand side can be evaluated by the inequality

By virtue of this, we have

The latter inequality leads us to the desired a priori estimate:

(11) |

The estimate (11) expresses the stability of the solution of the problem (9), (10) with respect to the initial data and the right-hand side and is quite similar to the estimate (7).

To solve numerically the problem (9), (10), we apply the implicit two-level scheme (Samarskii, 2001). Let be a step of the uniform grid in time so that , . For a constant weight parameter , we define

To approximate equation (9), let us consider the following two-level scheme:

(12) |

(13) |

For , the scheme (12), (13) becomes the explicit scheme, for , we get the fully implicit scheme, whereas corresponds to the symmetric scheme (the Crank-Nicolson scheme). The condition

is necessary and sufficient for the stability of the scheme (Samarskii, 2001; Samarskii et al., 2002). In particular, the following statement is true.

###### Theorem 1

[Proof.] Let us estimate the transition operator. Rewrite the scheme with weights (12) in the form

(15) |

where is the operator of transition to a new time level:

Let us formulate the restrictions on the weight that guarantee the following two-sided inequality:

(16) |

where . Taking into account the commutativity of the operators and , (16) is equivalent to

The right inequality is fulfilled for all . The left inequality gives

For non-negative operators , it always holds for . For (15), in view of (16), we get

which provides the estimate (14).

## 3 Two-component scheme based on a partition of unity

For the differential problem under the consideration, we select the domain decomposition

(17) |

with overlapping subdomains () (Quarteroni and Valli, 1999; Toselli and Widlund, 2005). To organize parallel computations, each subdomain consists of a set of disconnected subdomains.

To construct schemes of domain decomposition, we use a partition of unity for the computational domain (Laevsky, 1987; Mathew et al., 1998). Each separate subdomain we associate with the function such that

where

The standard approach (Samarskii et al., 2002) is based on the additive representation of the operator of the problem (9), (10):

(18) |

where each individual operator term is associated with the separate subdomain . For instance, in view of (1), it is natural to put

In this case, for the representation (18), we have

The construction and investigation of domain decomposition schemes for the unsteady problems (9), (10), (18) involves the consideration of the appropriate splitting schemes (Vabishchevich, 2014). In the case of two-component splitting, we can apply the following additive operator–difference schemes of ADI type: the Douglas–Rachford or Peaceman–Rachford scheme and factorized schemes, which generalize them.

In the factorized scheme, an approximate solution at a new time level is evaluated from the equation

(19) |

where

(20) |

For , the factorized scheme (19), (20) corresponds to Peaceman–Rachford scheme, whereas results in the Douglas–Rachford scheme.

For the numerical implementation of the factorized scheme, we can introduce the auxiliary value , which is determined from the equation

(21) |

For the approximate solution at a new time level, we have

(22) |

Thus, explicit-implicit approximations are used for in (9) taking into account the splitting (18).

Now we provide an estimate for the stability of the factorized scheme, which is similar to the estimate (14) for the scheme with weights (12), (13).

###### Theorem 2

[Proof.] The consideration is conducted by analogy with the proof of theorem 1. From (19), we have

(24) |

In the case of (19), (20), the transition operator can be written

where

Using Kellogg’s lemma (see, e.g., Kellogg (1964); Grossmann et al. (2007); Vabishchevich (2014)), we obtain

For , we have

In view of this, from (24), we get

This inequality leads to the estimate (23).

## 4 Domain decomposition scheme based on indicator functions of subdomains

A new variant of domain decomposition schemes is designed using indicator functions for the subdomains . Let us introduce

Define also the indicator function for the domain of overlap :

Thus, we have

(25) |

Operators of decomposition are constructed using the indicator functions of the subdomains and :

For equality ((25), we have a three-component representation of the problem operator:

(26) |

where

It is necessary to construct splitting schemes for the problem (9), (10), (26), where a transition to a new time level is performed, as before, by solving problems for the operators .

Taking into account the splitting (26), similarly to (21), (22), an approximate solution is obtained from equations

(27) |

(28) |

To study the stability of the scheme (27), (28), we employ theorem 2.

Suppose

In view of (26), we get

With the notation introduced above, the scheme (27), (28) is written as

(29) |

(30) |

where

Let

This allows to write (29), (30) in the form

(31) |

(32) |

where

To study the stability of the scheme (31), (32), we use theorem 2. Under the restriction , the following estimate holds:

where

This implies

(33) |

where

The main result of our consideration is the following basic statement on the stability of schemes of domain decomposition constructed on the basis of the indicator functions of the subdomains.

## 5 Numerical example

Numerical experiments presented here are of comparative nature. We consider the two-level scheme with weights (12), (13) as the reference scheme that provides the benchmark numerical solution for a comparison with the results of two schemes of domain decomposition with overlapping subdomains. Namely, these are the scheme based on the partition of unity for the domain (13), (18)–(20) and the scheme that uses the indicator functions of subdomains (13), (26)–(28). In our consideration, we monitor the proximity of the above decomposition schemes to the reference scheme. We confine ourselves to the case of schemes with .

Let be the benchmark numerical solution obtained using the implicit scheme (12), (13), whereas are the solutions from the decomposition scheme (13), (18)–(20) and (13), (26)–(28), respectively. We evaluate the deviation of the solution obtained using the domain decomposition methods from the benchmark solution derived without domain decomposition as the error:

The computational domain is divided by the variable into two subdomains with the overlap width . For the basic variant , the functions , which define the partition operators, are shown in Fig.1. The uniform spatial grid was used with piecewise-linear finite elements on triangles. The number of steps in time is .

The error of the domain decomposition schemes is presented in Fig.2. In the example considered here, the accuracy of the scheme based on the indicator functions of the subdomains (13), (26)–(28) is higher. The benchmark solution

at the final time moment is shown in Fig.

3. The deviation of the solutions obtained with two schemes of domain decomposition from the benchmark one is given in Fig.4, 5.Reducing the overlap width by half (see Fig.6) results in decreasing the accuracy of the approximate solution. For , both schemes under the consideration coincide and lead to the domain decomposition scheme with non-overlapping subdomains.

## 6 Conclusions

The boundary value problem for a second-order parabolic equation is considered. The standard finite element approximation in space is employed. For approximation in time, the two-level scheme with weight (-method) is used.

Iteration-free schemes of domain decomposition with overlapping subdomains are constructed on the basis of a partition of unity for a computational domain. In the case of a two-component domain decomposition, approximation in time is designed using factorized schemes, which generalize the standard ADI schemes.

A new class of domain decomposition schemes with overlapping subdomains based on indicator functions of subdomains is proposed for unsteady problems. In this case, for the problem operator, we have a three-component representation with decomposition operators connected with subdomains and their intersection. The unconditional stability of these schemes is established for two-component schemes of domain decomposition.

Numerical results for the model two-dimensional problem demonstrate the robustness of the new decomposition scheme with overlapping subdomains. A comparison of the conventional scheme based on a partition of unity with the scheme that used the indicator functions of the subdomains demonstrates its advantage in accuracy.

## Acknowledgements

This work was supported by the grant of Russian Federation Government (agreement # 14.Y26.31.0013).

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