Two-dimensional Elliptic Problems Research Articles

The singular function boundary integral method for 3-D Laplacian problems with a boundary straight edge singularity

Three-dimensional Laplace problems with a boundary straight-edge singularity caused by two intersecting flat planes are considered. The solution in the neighbourhood of the straight edge can be expressed as an asymptotic expansion involving the eigenpairs of the analogous two-dimensional problem in polar coordinates, which have as coefficients the so-called edge flux intensity functions (EFIFs). The EFIFs are functions of the axial coordinate, the higher derivatives of which appear in an inner infinite series in the expansion. The objective of this work is to extend the singular function boundary integral method (SFBIM), developed for two-dimensional elliptic problems with point boundary singularities [G.C. Georgiou, L. Olson, G. Smyrlis, A singular function boundary integral method for the Laplace equation, Commun. Numer. Methods Eng. 12 (1996) 127–134] for solving the above problem and directly extracting the EFIFs. Approximating the latter by either piecewise constant or linear polynomials eliminates the inner infinite series in the local expansion and allows the straightforward extension of the SFBIM. As in the case of two-dimensional problems, the solution is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing harmonic equation in the Galerkin sense. The resulting discretized equations are reduced to boundary integrals by means of the divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers. The values of the latter are calculated together with the coefficients of the EFIFs. The SFBIM is applied to a test problem exhibiting fast convergence of order k+1 (k being the order of the approximation of the EFIFs) in the L2-norm and leading to accurate estimates for the EFIFs.

Two-point Taylor approximations of the solutions of two-dimensional boundary value problems

We consider two-dimensional elliptic boundary value problems of the form L(U)=F in Ω⊂R2,Ω bounded and open, with a Dirichlet boundary condition U|∂Ω=H, where L is a second order linear differential operator whose coefficients, as well as the functions F and H are differentiable up to a certain degree. In a recent paper [C. Kesan, Taylor polynomial solutions of second order linear partial differential equations, Appl. Math. Comput. 152 (2004) 29–41], a matrix algorithm is introduced to compute the standard Taylor polynomial of the solution U at a certain point in Ω. We propose an alternative formulation of the problem based on a redefinition of the unknown U and the use of the standard Frobenius method that simplifies the computation of the Taylor coefficients of U. We also consider the use of a two-point Taylor representation of the solution, instead of the classical one-point Taylor representation, which gives a more uniform approximation of the solution.

Asymptotics of the solution of Laplace’s equation with mixed boundary conditions

A uniform asymptotic expansion of the solution of a two-dimensional elliptic problem with mixed boundary conditions is found. A physical application of the result is discussed.

A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

AbstractIn this article, we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions. Let Ω and ω be two bounded domains of Rd such that ω̅⊂Ω. For a linear elliptic problem in Ω\ω̅ with Robin boundary condition on the boundary ϒ of ω, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given; they suggest optimal order of convergence.

Multiscale mass conservative domain decomposition preconditioners for elliptic problems on irregular grids

Multiscale methods can in many cases be viewed as special types of domain decomposition preconditioners. The localisation approximations introduced within the multiscale framework are dependent upon both the heterogeneity of the reservoir and the structure of the computational grid. While previous works on multiscale control volume methods have focused on heterogeneous elliptic problems on regular Cartesian grids, we have tested the multiscale control volume formulations on two-dimensional elliptic problems involving heterogeneous media and irregular grid structures. Our study shows that the tangential flow approximation commonly used within multiscale methods is not suited for problems involving rough grids. We present a more robust mass conservative domain decomposition preconditioner for simulating flow in heterogeneous porous media on general grids.

A spectral fictitious domain method with internal forcing for solving elliptic PDEs

A fictitious domain method is presented for solving elliptic partial differential equations using Galerkin spectral approximation. The fictitious domain approach consists in immersing the original domain into a larger and geometrically simpler one in order to avoid the use of boundary fitted or unstructured meshes. In the present study, boundary constraints are enforced using Lagrange multipliers and the novel aspect is that the Lagrange multipliers are associated with smooth forcing functions, compactly supported inside the fictitious domain. This allows the accuracy of the spectral method to be preserved, unlike the classical discrete Lagrange multipliers method, in which the forcing is defined on the boundaries. In order to have a robust and efficient method, equations for the Lagrange multipliers are solved directly with an influence matrix technique. Using a Fourier–Chebyshev approximation, the high-order accuracy of the method is demonstrated on one- and two-dimensional elliptic problems of second- and fourth-order. The principle of the method is general and can be applied to solve elliptic problems using any high order variational approximation.

The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.

A Cartesian-Grid Discretisation Scheme Based on Local Integrated RBFNs for Two-Dimensional Elliptic Problems

This paper reports a new numerical scheme based on Cartesian grids and local integrated radial-basis-function networks (IRBFNs) for the solution of second-order elliptic differential problems defined on two-dimensional regular and irregular domains. At each grid point, only neighbouring nodes are activated to construct the IRBFN approximations. Local IRBFNs are introduced into two different schemes for discretisation of partial differential equations, namely point collocation and control-volume (CV)/subregion-collocation. Linear (e.g. heat flow) and nonlinear (e.g. lid-driven triangular-cavity fluid flow) problems are considered. Numerical results indicate that the local IRBFN CV scheme outperforms the local IRBFN point-collocation scheme regarding accuracy. Moreover, the former shows a similar level of the matrix condition number and a significant improvement in accuracy over a linear CV method.

Asymptotics for mixed Dirichlet–Robin problems in irregular domains

We study the asymptotic behaviour of the solutions of two-dimensional elliptic problems with Robin boundary conditions on the “prefractal” curves approximating the Koch curve type fractals.

On the geometric convergence of optimized Schwarz methods with applications to elliptic problems

The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Optimized Schwarz Methods use Robin conditions on the artificial interfaces for information exchange at each iteration, and for which one can optimize the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Optimized Schwarz Methods still lack a complete theory. In this paper, an abstract Hilbert space version of the Optimized Schwarz Method (OSM) is presented, together with an analysis of conditions for its geometric convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Optimized Schwarz Methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence factor ρ(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence factor does not appear to depend on h.

Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem

In this note, we consider the problem $$-\Delta u = u^p \quad {\rm{in}}\; \Omega, \quad u > 0 \quad {\rm{in}} \; \Omega, \quad u |_{\partial \Omega} = 0$$ on a smooth bounded domain Ω in \({\mathbb {R}^2}\) for p > 1. Let up be a positive solution of the above problem with Morse index less than or equal to \({m \in \mathbb {N}}\) . We prove that if up further satisfies the assumption \({p {\int_\Omega} |\nabla u_p|^2 dx = O(1)}\) as p → ∞, then the number of maximum points of up is less than or equal to m for p sufficiently large. If Ω is convex, we also show that a solution of Morse index one satisfying the above assumption has a unique critical point and the level sets are star-shaped for p sufficiently large.

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.

A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces

We study the numerical solution of a Cauchy problem for a self-adjoint elliptic partial differential equation uzz − Lu = 0 in three space dimensions (x, y, z), where the domain is cylindrical in z. Cauchy data are given on the lower boundary and the boundary values on the upper boundary are sought. The problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of the two-dimensional elliptic operator L (via its eigenfunction expansion), and it is shown that the solution is stabilized (regularized) if the large eigenvalues are cut off. We suggest a numerical procedure based on the rational Krylov method, where the solution is projected onto a subspace generated using the operator L−1. This means that in each Krylov step, a well-posed two-dimensional elliptic problem involving L is solved. Furthermore, the hyperbolic cosine is evaluated explicitly only for a small symmetric matrix. A stopping criterion for the Krylov recursion is suggested based on the relative change of an approximate residual, which can be computed very cheaply. Two numerical examples are given that demonstrate the accuracy of the method and the efficiency of the stopping criterion.

Convergence and superconvergence analysis of a new quadratic Hermite-type triangular element on anisotropic meshes

A new quadratic Hermite-type triangular finite element is conceived to solve a class of two-dimensional second-order elliptic boundary value problems. Its error estimates on anisotropic meshes are developed. Furthermore, we verify that some conditions set to the meshes contribute to the proof of its superconvergence properties, which can improve the approximation results. Numerical examples are given to confirm our theoretical analysis.

Reproducing polynomial (singularity) particle methods and adaptive meshless methods for two-dimensional elliptic boundary value problems

Oh et al. [H.-S. Oh, J.G. Kim, J.W. Jeong, The closed form reproducing polynomial particle shape functions for meshfree particle methods, Comput. Methods Appl. Mech. Engrg. 196 (2007) 3435–3461] introduced the reproducing polynomial particle (RPP) shape functions that are piecewise polynomial and satisfy the Kronecker delta property. In this paper, we introduce reproducing polynomial particle methods (RPPM) that is the Galerkin approximation method associated with the use of the RPP approximation space. Planting particles in the computation domain in a patchwise uniform manner, we also introduce patchwise RPPM. Furthermore, constructing partition of unity functions with flat-top adaptively, we introduce adaptive patchwise RPPM and compare it with reproducing singularity particle methods (RSPM) that is the Galerkin approximation method associated with the use of reproducing singularity particle (RSP) shape functions.

A multidomain integrated‐radial‐basis‐function collocation method for elliptic problems

AbstractThis article is concerned with the use of integrated radial‐basis‐function networks (IRBFNs) and nonoverlapping domain decompositions (DDs) for numerically solving one‐ and two‐dimensional elliptic problems. A substructuring technique is adopted, where subproblems are discretized by means of one‐dimensional IRBFNs. A distinguishing feature of the present DD technique is that the continuity of the RBF solution across the interfaces is enforced with one order higher than with conventional DD techniques. Several test problems governed by second‐ and fourth‐order differential equations are considered to investigate the accuracy of the proposed technique. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Multilevel Preconditioning of Two-dimensional Elliptic Problems Discretized by a Class of Discontinuous Galerkin Methods

We present optimal-order preconditioners for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems. We consider two variants of hierarchical splittings where the underlying finite element meshes are geometrically nested. A specific assembling process is proposed which facilitates the local analysis of the angle between the related subspaces. By applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated that can be associated with a hierarchy of coarse versions of DG approximations of the original problem. New bounds for the constant $\gamma$ in the strengthened Cauchy-Bunyakowski-Schwarz inequality are derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach.

Superconvergence Analysis of a Nonconforming Triangular Element on Anisotropic Meshes

The class of anisotropic meshes we conceived abandons the regular assumption. Some distinct properties of Carey’s element are used to deal with the superconvergence for a class of two-dimensional second-order elliptic boundary value problems on anisotropic meshes. The optimal results are obtained and numerical examples are given to confirm our theoretical analysis.

Three-step BLAGE iterative method for two-dimensional elliptic boundary value problems with singularity†

We present a new three-step block alternating group explicit (BLAGE) algorithm for solving block symmetric linear systems derived from both five-point second-order and nine-point fourth-order discretizations to solve the two-dimensional elliptic equation in both rectangular and cylindrical polar coordinates. The three-step BLAGE algorithm should save time compared with the two-step BLAGE algorithm as the aim of the procedure is to eliminate the evaluation of two similar terms which appear in the algorithm. The convergence theory of the algorithm is reported briefly. Numerical results obtained from three test problems are presented to demonstrate the utility of the proposed three-step BLAGE method compared with the corresponding two-step BLAGE iterative method and the block successive over-relaxation (BSOR) iterative method. † Dedicated to the late Professor D.J. Evans

A second-order continuity domain decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems

This paper presents a second-order continuity non-overlapping domain decomposition (DD) technique for numerically solving second-order elliptic problems in two-dimensional space. The proposed DD technique uses integrated Chebyshev polynomials to represent the solution in subdomains. The constants of integration are utilized to impose continuity of the second-order normal derivative of the solution at the interior points of subdomain interfaces. To also achieve a C 2 function at the intersection of interfaces, two additional unknowns are introduced at each intersection point. Numerical results show that the present DD method yields a higher level of accuracy than conventional DD techniques based on differentiated Chebyshev polynomials.