Order Elliptic Partial Differential Equations Research Articles

Numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations (PDEs). Various discretisation procedures such as the finite difference method result in a problem of solving large, sparse systems of linear equations. In this paper, a group iterative numerical scheme based on the rotated (skewed) five-point finite difference discretisation is proposed for the solution of a fourth order elliptic PDE which represents physical situations in fluid mechanics and elasticity. The rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain. We describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using Message-Passing Interface (MPI) programming environment. A comparative study with another group iterative scheme derived from the centred difference formula is also presented. A detailed performance analysis of the parallel implementations of both group methods will be reported and discussed.

Goal-Oriented Adaptivity in Pointwise State Constrained Optimal Control of Partial Differential Equations

We derive primal-dual weighted goal-oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations. The constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. In the case of sufficiently regular active (or coincidence) sets and problem data, a further decomposition of the multiplier into a regular $L^2$-part on the active set and a singular part concentrated on the boundary between the active and inactive set allows us to further characterize the mismatch error. The paper ends with a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of only smooth curves or points.

Maximum norm error estimates for Div least-squares method for Darcy flows

Least-squares finite element methods for second order elliptic partial differential equations such as Darcy flows are considered. While there has been a significant progress in terms of obtaining error estimates for the methods, the estimates are essentially based on $L_2$-norm of the error. In this paper, we provide maximum norm error estimates for the primary variable using a smoothed Green's function introduced in [33] and maximum norm error for the dual variables by taking advantage of the fact that least-squares solutions are higher-order perturbations of Galerkin solutions [8].

Steklov–Neumann eigenproblems and nonlinear elliptic equations with nonlinear boundary conditions

We study the solvability of nonlinear second order elliptic partial differential equations with nonlinear boundary conditions. We introduce the notion of “eigenvalue-lines” in the plane; these eigenvalue-lines join each Steklov eigenvalue to the first eigenvalue of the Neumann problem with homogeneous boundary condition. We prove existence results when the nonlinearities involved asymptotically stay, in some sense, below the first eigenvalue-lines or in a quadrilateral region (depicted in Fig. 1) enclosed by two consecutive eigenvalue-lines. As a special case we derive the so-called nonresonance results below the first Steklov eigenvalue as well as between two consecutive Steklov eigenvalues. The case in which the eigenvalue-lines join each Neumann eigenvalue to the first Steklov eigenvalue is also considered. Our method of proof is variational and relies mainly on minimax methods in critical point theory.

Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We develop index theories for linear selfadjoint operator equations and investigate multiple solutions for asymptotically linear operator equations. The operator equations consist of two kinds: the first has finite Morse index and can be used to investigate second order Hamiltonian systems and elliptic partial differential equations; the second may have infinite Morse index and can be used to investigate first order Hamiltonian systems.

On the solvability of the higher order elliptic boundary value problem

In this paper, by employing a non-variational version of a max–min principle, new results on existence and uniqueness of generalized solution to the boundary value problem of higher order elliptic partial differential equations at resonance are given.

On some new approximate factorization methods for block tridiagonal matrices suitable for vector and parallel processors

In this paper, to obtain an efficient parallel algorithm to solve sparse block-tridiagonal linear systems, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are suitable when the desired goal is to maximize parallelism. Moreover, some theoretical results concerning these preconditioners are presented and how to construct preconditioners effectively for any nonsingular block tridiagonal H-matrices is also described. In addition, the validity of these preconditioners is illustrated with some numerical experiments arising from the second order elliptic partial differential equations and oil reservoir simulations.

An additive Schwarz preconditioner for the mortar-type rotated [formula omitted] FEM for elliptic problems with discontinuous coefficients

In this paper, we propose an additive Schwarz preconditioner for the mortar-type rotated Q 1 finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. The work here is an extension of the research presented in [L. Marcinkowski, Additive Schwarz method for mortar discretization of elliptic problems with P 1 non-conforming elements, BIT 45 (2005) 375–394]. Our analysis is valid for rectangular or L-shaped domains, which are partitioned by rectangular subdomains and meshes. We have shown that our proposed method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is O ( ( 1 + log ( H / h ) ) 2 ) , which is independent of the jump in the coefficient. Numerical experiments presented in this paper have confirmed our theoretical analysis.

On reproducing uniform flow exactly on general hexahedral cells using one degree of freedom per surface

The solution of second order elliptic partial differential equations typically arising for flow problems, e.g., in porous media, is frequently expressed as face fluxes of a flow field at hexahedral cells. In this letter, we show by example that the following properties are incompatible: (1) A local velocity reconstruction, (2) reproduction of uniform flow for general hexahedral cells, and (3) a flow field in H(div). This is in particular relevant to mixed finite element methods approximating H(div), and to other methods trying to reconstruct a flow field in a hexahedron using only local face fluxes, e.g., streamline tracing methods for reservoir simulation.

A finite volume method on general surfaces and its error estimates

In this paper, we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface. The discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons. The optimal error estimates of the approximate solution are proved in both the H 1 and L 2 norms which are of first order and second order respectively under mesh regularity assumptions. Some numerical tests are also carried out to experimentally verify our theoretical analysis.

Parallel MIC(0) preconditioning of 3D elliptic problems discretized by Rannacher–Turek finite elements

Novel parallel algorithms for the solution of large FEM linear systems arising from second order elliptic partial differential equations in 3D are presented. The problem is discretized by rotated trilinear nonconforming Rannacher–Turek finite elements. The resulting symmetric positive definite system of equations A x = f is solved by the preconditioned conjugate gradient algorithm. The preconditioners employed are obtained by the modified incomplete Cholesky factorization MIC(0) of two kinds of auxiliary matrices B that both are constructed as locally optimal approximations of A in the class of M -matrices. Uniform estimates for the condition number κ ( B − 1 A ) are derived. Two parallel algorithms based on the different block structures of the related matrices B are studied. The numerical tests confirm theory in that the algorithm scales as O ( N 7 / 6 ) in the matrix order N .

An adaptive hp-version of the multilevel particle–partition of unity method

This paper is concerned with the hp-adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle–partition of unity method. The proposed refinement scheme automatically constructs new discretization points (or particles), the meshfree analogue of an adaptive h-refinement, and local approximation spaces with better local resolution, a p-refinement. The refinement process is steered with the help of an a-posteriori subdomain-type error indicator. We present results of numerical experiments in two and three space dimensions which demonstrated the overall efficiency of the proposed scheme.

On geometric variational models for inpainting surface holes

Geometric approaches for filling-in surface holes are introduced and studied in this paper. The basic principle is to choose the completing surface as one which minimizes a power of the mean curvature. We interpret this principle in a level set formulation, that is, we represent the surface of interest in implicit form and we construct an energy functional for the embedding function u. We first explore two different formulations (which can be considered as alternative) inspired by the above principle: in the first one we write the mean curvature as the divergence of the normal vector field θ to the isosurfaces of u; in the second one we used the signed distance function D to the surface as embedding function and we write the mean curvature in terms of it. Then we solve the Euler–Lagrange equations of these functionals which consist of a system of second order partial differential equations (PDEs) for u and θ , in the first case, or a fourth order PDE for D in the second case. Then, simpler methods based on second order elliptic PDEs, like Laplace equation or the absolutely minimizing Lipschitz extension, are also proposed and compared with the above higher order methods. The theoretical and computational framework, as well as examples with synthetic and real data, are presented in this paper.

Goal-Oriented Adaptivity in Control Constrained Optimal Control of Partial Differential Equations

Dual-weighted goal-oriented error estimates for a class of pointwise control constrained optimal control problems for second order elliptic partial differential equations are derived. It is demonstrated that the constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. The paper also contains a posteriori error estimators for the $L^2$-norm of the error in the state and in the adjoint state.

Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions

We describe several topics within the theory of linear and nonlinear second order elliptic Partial Differential Equations. Through elementary approaches, we first explain how elliptic and parabolic PDEs are related to central issues in Probability and Geometry. This leads to several concrete equations. We classify them and describe their regularity theories. After this, most of the paper focuses on the ABP technique and its applications to the classical isoperimetric problem for which we present a new original proof, the symmetry result of Gidas-Ni-Nirenberg, and the regularity theory for fully nonlinear elliptic equations.

Conjugate gradient-boundary element solution using multiple reciprocity method for distributed elliptic optimal control problems

This paper deals with the numerical solution of optimal control problems, where the state equations are given by the fourth order elliptic partial differential equations. An iterative algorithm for this class of problems is developed. This new proposal is obtained by combining the Conjugate Gradient Method (CGM) with the Boundary Element Method (BEM) and Multiple Reciprocity Method (MRM). The local error estimates based on the stability of this scheme in the H 2 norm, L 2 norm and L ∞ norm are obtained. Finally, the numerical results on a test case show that this method is correct and feasible.

On discontinuous Galerkin method and semiregular family of triangulations

Discretization of second order elliptic partial differential equations by discontinuous Galerkin method often results in numerical schemes with penalties. In this paper we analyze these penalized schemes in the context of quite general triangular meshes satisfying only a semiregularity assumption. A new (modified) penalty term is presented and theoretical properties are proven together with illustrative numerical results.

FETI Domain Decomposition Methods for Second Order Elliptic Partial Differential Equations

AbstractA survey on certain Finite Element Tearing and Interconnecting (FETI) methods for second order elliptic partial differential equations is given. FETI methods are nonoverlapping domain decomposition algorithms where continuity across subdomain boundaries is treated, sometimes only in parts, by Lagrange multipliers. The family of FETI algorithms is among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. In this article, an overview is given of the classical one‐level FETI method and the more recent dual‐primal FETI methods; both type of algorithms are considered for scalar, second order, elliptic equations and the system of linear elasticity, and are illustrated by some computational results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimal Alternating Direction Implicit Preconditioners for Conjugate Gradient methods

In this work the (Extrapolated) Alternating Direction Implicit ((E)ADI) Preconditioners for the Conjugate Gradient (CG) Method are applied for the solution of the second order elliptic partial differential equation (PDE) in a rectangle under Dirichlet boundary conditions. It is shown that the spectral condition number for the discretized model problem, when the ADI-CG Method is used, is an order of magnitude smaller than that of the CG Method. The PDE is approximated by uniform 5- and 9-point stencils and it is shown that the determination of the optimal condition number is equivalent to that of the determination of the optimal EADI scheme. Analytic expressions for the optimal acceleration and extrapolation parameters are obtained in both cases of the 5- and 9-point schemes; those for the 5-point complete known ones while those for the 9-point are novel. Finally, numerical examples show that the EADI preconditioners proposed make the CG methods compete well against most of the fastest well-known methods.

ON THE EXISTENCE OF A SOLUTION IN WEIGHTED SOBOLEV SPACE TO THE RIEMANN–HILBERT PROBLEM FOR AN ELLIPTIC SYSTEM WITH PIECEWISE CONTINUOUS BOUNDARY DATA

Given a first order elliptic partial differential equation we construct a solution which solves a given Riemann–Hilbert boundary value problem whose coefficients have singularities of the first kind at a finite number of some prescribed isolated points and are Holder–continuous outside those points while the free term has a finite number of integrable power singularities at some prescribed points. It is shown that the solution belongs to some weighted Sobolev space $W_{1, p}(D; \rho)$ , where the weight function $\rho = \rho(z; \partial D)$ is the distance of the variable point $z$ from the boundary $\partial D$ raised to a certain power. The problem is solved by first reducing it to an analogous problem for holomorphic functions. The latter is then solved.