Iterative algorithms of domain decomposition for the solution of a quasilinear elliptic problem

This paper deals with iterative algorithms for domain decomposition applied to the solution of a quasilinear elliptic problem. Two iterative algorithms are examined: the first one is the Schwarz alternating procedure and the second algorithm is suitable for parallel computing. Convergence results are established in the two-domain and multidomain decomposition cases. Some issues of parallel implementation of these algorithms are discussed.     

Spectral domain embedding for elliptic PDEs in complex domains

Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such as rectangles or disks. Domain decomposition methods/spectral element methods extend the applicability of spectral methods to more complex geometries. An alternative is to embed the irregular domain into a regular one. This paper uses the spectral method with domain embedding to solve PDEs on complex geometry. The running time of the new algorithm has the same order as that for the usual spectral collocation method for PDEs on regular geometry. The algorithm is extremely simple and can handle Dirichlet, Neumann boundary conditions as well as nonlinear equations.     

A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

Balancing domain decomposition for problems with large jumps in coefficients

The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the first-named author is used to develop condition number bounds for conforming linear elements in two and three dimensions. The bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains, and grow only as the squared logarithm of the mesh size . Computational experiments for two- and three-dimensional problems confirm the theory.

     

A domain decomposition method on nested domains and nonmatching grids

A one directionally coupled problem on two nested domains is analyzed. The global domain and the subdomain are discretized by two triangulations that do not match on the subdomain. The connection between the two grids is established by using a stable projection operator onto the interface. An a priori error analysis is carried out and several numerical examples are given. The method is ideally suited for the case of a moving subdomain. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 374–387, 2004.     

Adaptive wavelet frame domain decomposition methods for nonlinear elliptic equations

In this article, we are concerned with the numerical treatment of nonlinear elliptic boundary value problems. Our method of choice is a domain decomposition strategy. Partially following the lines from (Cohen, Dahmen and deVore, SIAM J Numer Anal 41 (2003), 1785–1823; Kappei, Appl Anal J Sci 90 (2011), 1323–1353; Lui, SIAM J Sci Comput 21 (2000), 1506–1523; Stevenson and Werner, Math Comp 78 (2009), 619–644), we develop an adaptive additive Schwarz method using wavelet frames. We show that the method converges with an asymptotically optimal rate and support our theoretical results with numerical tests in one and two space dimensions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A Schwarz-based domain decomposition method for the dispersion equation

We propose a Schwarz-based domain decomposition method for solving a dispersion equation consisting on the linearized KdV equation without the advective term, using simple interface operators based on the exact transparent boundary conditions for this equation. An optimization process is performed for obtaining the approximation that provides the method with the fastest convergence to the solution of the monodomain problem.     

Positive solutions for a class of multiparameter ordinary elliptic systems

Using a fixed point theorem due to M.A. Krasnosel'skii, the upper-lower solutions method and fixed point index theory, we study existence, non-existence and multiplicity of positive solutions for a class of systems of second-order ordinary differential equations. We give a new notion of superlinearity for nonlinearities. Observe that the nonlinearities may even be, in some sense, singular. As an application, we show multiplicity results for a class of semilinear elliptic systems in both bounded annular domains and exterior domains. In addition, we apply our results to fourth-order boundary value problems.     

An index formula for elliptic systems in the plane

An index formula is proved for elliptic systems of P.D.E.'s with boundary values in a simply connected region in the plane. Let denote the elliptic operator and the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function defined on the unit cotangent bundle of was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function have invertible values. In the present paper, the index of is expressed in terms of the winding number of the determinant of .

     

Short interval asymptotics for a class of arithmetic functions

Summary We provide a general asymptotic formula which permits applications to sums like <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sum_{x< n\le x+y} \big(d(n)\big)^2, \quad \sum_{x< n\le x+y} d(n^3),\quad \sum_{x< n\le x+y}\big(r(n)\big)^2, \quad \sum_{x< n\le x+y}r(n^3), $$ where $d(n)$ and $r(n)$ are the usual arithmetic functions (number of divisors, sums of two squares), and $y$ is small compared to~$x$.     

A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems

In recent years, competitive domain-decomposed preconditioned iterative techniques of Krylov-Schwarz type have been developed for nonsymmetric linear elliptic systems. Such systems arise when convection-diffusion-reaction problems from computational fluid dynamics or heat and mass transfer are linearized for iterative solution. Through domain decomposition, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow effective solution on parallel machines. A central question is how to choose these small problems and how to arrange the order of their solution. Different specifications of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILU-family of preconditioners, on some nonsymmetric or indefinite elliptic model problems discretized by finite difference methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.     

Numerical results for positive solutions of a superlinear elliptic equation

In this work we present a numerical approach for finding positive solutions of the type −Δu = λf(u) for x  Ω, with Dirichlet boundary condition, where f is a superlinear function of u. We will show in which range of λ, this problem achieves multiple numerical solutions and what is the behavior of the branches of solutions.     

Dirichlet boundary conditions for elliptic operators with unbounded drift

We study the realisation of the operator in with Dirichlet boundary condition, where is a possibly unbounded open set in , is a semi-convex function and the measure lets be formally self-adjoint. The main result is that at is a dissipative self-adjoint operator in .

     

Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds II

Extending our recent work for the semilinear elliptic equation on Lipschitz domains, we study a general second-order Dirichlet problem in . We improve our previous results by studying more general nonlinear terms with polynomial (and in some cases exponential) growth in the variable . We also study the case of nonnegative solutions.

     

An efficient domain decomposition method for three-dimensional parabolic problems

A non-overlapping domain decomposition algorithm to solve three-dimensional parabolic partial differential equations is presented. It has been shown in this paper that the algorithm is unconditionally stable and efficient. Spectral radii for the interface and interior region are provided. Unlike two-dimensional problem, it has been found out that estimating the values of the points of the interface in three-dimensional problem is no longer negligible.     

Liouville type theorems for fourth order elliptic equations in a half plane

Consider an elliptic equation in the half plane with boundary conditions if and if where are second and third order differential operators. It is proved that if and, for some , if if where for some nonnegative integer , then . Results of this type are also established in case under different conditions on and ; furthermore, in one case has a lower order term which depends nonlocally on . Such Liouville type theorems arise in the study of coating flow; in fact, they play a crucial role in the analysis of the linearized version of this problem. The methods developed in this paper are entirely different for the two cases (i) and (ii) ; both methods can be extended to other linear elliptic boundary value problems in a half plane.

     

An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations

The time-harmonic Maxwell equations are considered in the low-frequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields.

     

The de Rham complex on unbounded domains with Sobolev space topology

This paper studies the operator dd^∗+d^∗d acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and d^∗ is the adjoint of d calculated using the Sobolev space topology. The domain of d^∗ is determined and an expression for d^∗ is obtained. The operator dd^∗+d^∗d gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators.