Parallel Performance of an MPI Solver for 3D Elasticity Problems

On existence and uniqueness of solutions of boundary value problems of fourth order elliptic partial differential equations with variable coefficients

We compare a block successive overrelaxation method (BSOR) with a direct method for solving difference equations derived from fourth order elliptic partial differential equations. The direct method appears to be competitive in time and accuracy for a general problem.

In this paper, we propose a BDDC preconditioner for the mortar-type rotated finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. We construct an auxiliary discrete space and build our algorithm on an equivalent auxiliary problem, and we present the BDDC preconditioner based on this constructed discrete space. Meanwhile, in the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is independent of the jumps of the coefficients, and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis. MSC:65N55, 65N30.

We consider second order elliptic partial differential equations with coefficients that are singular or degenerate at an interior point of the domain. This paper presents formulation and analysis of a novel weighted-norm least squares finite element method for this class of problems. We propose a weighting scheme that eliminates the pollution effect and recovers optimal convergence rates. Theoretical results are carried out in appropriately weighted Sobolev spaces and include ellipticity bounds on the weighted homogeneous least squares functional, regularity bounds on the elliptic operator, and error estimates. Numerical experiments confirm the predicted error bounds.

A classical problem in materials science involves the determination of effective transport properties in composites. The simplest problem is perhaps described by the second order elliptic partial differential equation where σ is piecewise constant, supplemented by suitable boundary conditions or by a radiation condition at infinity. This is a standard model for steady-state heat conduction and conductivity. For our purposes, it will be more convenient to rewrite (1) as an interface problem; we seek a potential function u which is harmonic in each component, continuous, and satisfies the continuity of flux condition

The strong maximum principle revisited

The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with C1,1 boundary. We assume that at least one of the material parameters is W1,p for some p > 3. Using regularity theory for second order elliptic partial differential equations, we derive W1,p estimates and Hölder estimates for electric and magnetic fields up to the boundary, together with their higher regularity counterparts. We also derive interior estimates in bianisotropic media.

Second order elliptic partial differential equations arise in many important applications, including flow through porous media, heat conduction, the distribution of electrical or magnetic potential. The prototype is the Laplace problem, which in discrete form produces a coefficient matrix that is relatively easy to solve in a regular domain. However, the presence of anisotropy produces a matrix whose condition number is increased, making the resulting linear system more difficult to solve. In this work, we take the anisotropy into account in the discretization by mapping each anisotropic region into a ``stretched`` coordinate space in which the anisotropy is removed. The region is then uniformly triangulated, and the resulting triangulation mapped back to the original space. The effect is to generate long slender triangles that are oriented in the direction of ``preferred flow.`` Slender triangles are generally regarded as numerically undesirable since they tend to cause poor conditioning; however, our triangulation has the effect of producing effective isotropy, thus improving the condition number of the resulting coefficient matrix.

The transport of intensity equation (TIE) is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. However, the boundary conditions are difficult to obtain in practice. The fast Fourier transform (FFT) based TIE solutions are widely adopted for its speed and simplicity. However, it implies periodic boundary conditions, which lead to significant boundary artifacts when the imposed assumption is violated. In this work, TIE phase retrieval is considered as an inhomogeneous Neumann boundary value problem with the boundary values experimentally measurable around a hard-edged aperture, without any assumption or prior knowledge about the test object and the setup. The analytic integral solution via Green's function is given, as well as a fast numerical implementation for a rectangular region using the discrete cosine transform. This approach is applicable for the case of non-uniform intensity distribution with no extra effort to extract the boundary values from the intensity derivative signals. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.

A method for the numerical solution of the time dependent two-dimensional magneto-plasma dynamic coupled problem in the low magnetic Reynolds number case, is presented. The method is based on a time dependent two-dimensional description of the fluid dynamics iteratively coupled to a steady state two-dimensional description of the electrodynamics. This is allowed as the characteristic times of variation of the electrodynamic quantities are much lower than those of the fluid dynamic quantities. The fluid is assumed to be compressible and viscous, and governed by the Navier-Stokes equations. A semi-implicit finite difference pressure scheme has been utilised for the solution of the fluid dynamic problem. The electrodynamics is described by a second order elliptic partial differential equation obtained from the Maxwell equations and generalised Ohm's law. The elliptic equation is solved by means of an exponential fitting finite difference method. In order to test the algorithm described above, the transient of a plasma flow in a channel, caused by the variation of the externally applied magnetic field, has been studied.

Recent research has identified Adini’s rectangular element as an efficient higher order element for solving second order elliptic partial differential equations such as Poisson’s equation, which governs the steady state heat conduction problem. This type of element includes both the primary field variable and its spatial derivatives as nodal degrees of freedom. Compared to the conventional cubic elements of the Serendipity and Lagrange families, Adini’s element includes the minimum number of nodes per element and has the advantage that the nodal values of the spatial derivatives of the temperature field are directly retrieved from the FEM solution. As a result, the differentiation and averaging procedures that are typically used to obtain the nodal values of the temperature gradients are avoided. In this paper a generalized version of Adini’s element for solving two-dimensional steady state heat transfer problems in non-rectangular geometries is presented. Also, the traditional finite element formulation is modified to allow the application of essential boundary conditions without having to constrain the nodal values of the tangential derivative of the temperature. The resulting higher order element and modified FEM formulation are used to solve an example problem and the accuracy of the solution is compared with solutions obtained using the traditional linear, quadratic, and cubic Serendipity elements to show the efficiency, in terms of accuracy per number of degrees of freedom, of the proposed approach for finding the nodal values of the temperature gradients, which are required to compute the nodal values of the heat flux vector.

We propose a smooth fictitious domain/multiresolution method for enhancing the accuracy order in solving second order elliptic partial differential equations on general bivariate domains. We prove the existence and uniqueness of the solution of a corresponding discrete problem and a so-called interior error estimate which justifies the improved accuracy order. Numerical experiments are conducted on a Cassini oval.

Чехословацкий математический журнал, т. 10 (85) I960, Прага О РЕШЕНИЯХ ЭЛЛИПТИЧЕСКИХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ В ЧАСТНЫХ ПРОИЗВОДНЫХ ВТОРОГО ПОРЯДКА С НЕОГРАНИЧЕННЫМ ИНТЕГРАЛОМ ДИРИХЛЕ ИНДРЖИХ НЕЧАС (Jindfich Necas), Прага (Поступило в редакцию 19/V 1959 г.) В настоящей работе доказывается существование и единственность решения задачи Дирихле для эллиптического оператора второго порядка при условии, что краевое условие интегрируемо с квадратом.Граница области может быть довольно нерегулярной.1. ВВЕДЕНИЕ Подобно тому, как мы доказали теорему 4.4, можно доказать и Теорему 6.4.Пусть для любой непрерывной функции g существует одно и только одно классическое решение и проблемы Дирихле.Пусть это реше ние типа Винера.Тогда и = Вд.Литература

: Analytical development of a set of second order elliptic partial differential equations for the generation of three-dimensional curvilinear coordinates between two arbitrary shaped bodies is presented. The resulting equations have only two independent variables and therefore require an order of magnitude less working core capacity than when equations depending on all three independent variables are considered. The method also allows, in a straight forward manner the possibility of coordinate contraction in the desired regions. (Author)

Optimal Alternating Direction Implicit Preconditioners for Conjugate Gradient methods