A NOTE ON ELLIPTIC EQUATIONS WITH BMO COEFFICIENTS: REGULARITY THEORY

Wide sense stationary processes are a mainstay of classical signal processing. It is well known that they can be obtained by solving ordinary differential equations with constant coefficients whose right-hand side is a white noise. This paper addresses the extension of this construction to random fields defined on compact Lie groups. On an underlying compact Lie group, the paper studies left invariant second-order elliptic partial differential equations whose right-hand side is a spatial white noise. Quite often, the solution of a partial differential equation is not defined as a function but as a distribution. To adapt to this situation, the paper introduces a definition of wide sense stationary distributions on a compact Lie group. This is shown to be consistent with the more restricted definition of wide sense stationary fields given in a classic paper by Yaglom. It is proved that the solution of a partial differential equation, of the kind being studied, is a wide sense stationary distribution whose covariance structure is determined by the fundamental solution of the equation. As a concrete example, this paper describes the fundamental solution of the Helmholtz equation on the rotation group and the resulting covariance structure.

This paper is concerned with continuous and discrete approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form. The continuous approximation of these equations is achieved through the Vanishing Moment Method (VMM) which adds a small biharmonic term to the PDE. The structure of the new fourth-order PDE is a natural fit for Galerkin-type methods unlike the original second order equation since the highest order term is in divergence form. The well-posedness of the weak form of the perturbed fourth order equation is shown as well as error estimates for approximating the strong solution of the original second-order PDE. A $C^1$ finite element method is then proposed for the fourth order equation, and its existence and uniqueness of solutions as well as optimal error estimates in the $H^2$ norm are shown. Lastly, numerical tests are given to show the validity of the method.

Numerical solution of second-order nonlinear partial differential equations originating from physical phenomena using Hermite based block methods

Local properties of solutions of a second-order elliptic partial differential equation in two variables with a point singularity are studied. Existence of solutions, regularity and maximum principle are considered.

The properties of mappings by the solutions of second-order elliptic partial differential equations in the plane are studied. We obtain conditions on a function, continuous on the unit circle, that are sufficient for the solution of the Dirichlet problem in the open unit disk for the given equation with the given boundary function to be a homeomorphism between the open unit disk and a Jordan simply connected domain. The properties of the zeros of the solutions of the given equations are also studied. In particular, an analog of the main theorem of algebra is proved for polynomial solutions.

PROPERTIES OF SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS USING FINITE ELEMENT METHODS

A priori bounds are derived for the discrete solution of second-order elliptic partial differential equations (PDEs). The bounds have two contributions. First, the influence of boundary conditions is taken into account through a discrete maximum principle. Second, the contribution of the source field is evaluated in a fashion similar to that used in the treatment of the continuous a priori operators. Closed form expressions are, in particular, obtained for the case of a conservative, second-order finite difference approximation of the diffusion equation with variable scalar diffusivity. The bounds are then incorporated into a resilient domain decomposition framework, in order to verify the admissibility of local PDE solutions. The computations demonstrate that the bounds are able to detect most system faults, and thus considerably enhance the resilience and the overall performance of the solver.

On comparing optimum Alternating Direction Preconditioning and Extrapolated Alternating Direction Implicit schemes

A Continuous Five-step Implicit Block Unification Method for Numerical Solution of Second-Order Elliptic Partial Differential Equations (PDEs)

A mixed finite element method is developed to approximate the solution of a quasilinear second-order elliptic partial differential equation. The existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived.

An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analyzed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (𝒫p-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a 𝒫p-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem.

A continuous implicit block unification method (CIBUM) is developed through the interpolation and collocation approach using Hermite polynomial as the basis function. The method is chosen within the interval of step-number of five. The basis function was interpolated at the first two consecutive points while the collocation was done at all the points within the interval of integration. The discrete scheme and their corresponding first derivative were combined to form the five-step implicit block unification method (FIBUM) of order six. The FIBUM is applied to solve second-order elliptic partial differential equations via the method of lines by transforming the PDEs into ODEs. The basic properties of FIBUM were investigated and found to be convergence and p-stable. The method was implemented on five test problems varying from linear, nonlinear, and nonlinear Klein-Gordon differential equations, and the results were presented. The results established the accuracy of the FIBUM over the existing ones.

Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions

A mixed finite element method is developed to approximate the solution of a quasilinear second-order elliptic partial differential equation. The existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived.

SUMMARY A numerical method for the computation of the generalized flux/stress intensity factors (GFIFs/GSIFs) for the asymptotic solution of linear second-order elliptic partial differential equations in two dimensions in the vicinity of singular points is described. Special attention is given to heat transfer and elasticity problems. The singularities may be caused by re-entrant corners and abrupt changes in material properties. Such singularities are of great interest from the point of view of failure initiation: The eigenpairs, computed in a companion paper,' characterize the straining modes and their amplitudes (the GFIFs/GSIFs) quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the GFIFs/GSIFs. This paper addresses a general method based on the complementary weak formulation for determining the GFIFs/GSIFs numerically as a post-solution operation on the finite element solution vector. Importantly, the method is applicable to anisotropic materials, multi-material interfaces, and cases where the singularities are characterized by complex eigenpairs. An error analysis is sketched and numerical examples are presented to illustrate the effectiveness of the technique.