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"Boundary value problems."
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Elliptic Theory for Sets with Higher Co-dimensional Boundaries
Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher
co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields
a notion analogous to that of the harmonic measure, for sets of codimension higher than 1.
To this end, we turn to degenerate
elliptic equations. Let
In another article to appear, we will prove that when
eBook
FINITE ELEMENT APPROXIMATIONS FOR STOKES—DARCY FLOW WITH BEAVERS—JOSEPH INTERFACE CONDITIONS
Numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits. Such situations arise, for example, for groundwater flows in karst aquifers. The coupled flow is modeled by the Darcy equation in a porous medium and the Stokes equations in the conduit domain. On the interface between the matrix and conduit, Beavers—Joseph interface conditions, instead of the simplified Beavers—Joseph—Saffman conditions, are imposed. Convergence and error estimates for finite element approximations are obtained. Numerical experiments illustrate the validity of the theoretical results.
Journal Article
p4est : Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees
(ProQuest: ... denotes formulae/symbols omitted.)The authors present scalable algorithms for parallel adaptive mesh refinement and coarsening (AMR), partitioning, and 2:1 balancing on computational domains composed of multiple connected two-dimensional quadtrees or three-dimensional octrees, referred to as a forest of octrees. By distributing the union of octants from all octrees in parallel, they combine the high scalability proven previously for adaptive single-octree algorithms with the geometric flexibility that can be achieved by arbitrarily connected hexahedral macromeshes, in which each macroelement is the root of an adapted octree. A key concept of their approach is an encoding scheme of the interoctree connectivity that permits arbitrary relative orientations between octrees. They demonstrate the parallel scalability of p4est on its own and in combination with two geophysics codes. Using p4est they generate and adapt multioctree meshes with up to 5.13 x ... octants on as many as 220,320 CPU cores and execute the 2:1 balance algorithm in less than 10 seconds per million octants per process.
Journal Article
Three Boundary Value Problems for Complex Partial Differential Equations in the Lens Domain
In this paper, we investigate some boundary value problems for the Cauchy–Riemann equations in the lens domain
M
. We apply the parqueting-reflection method for the domain to achieve the points of the complex plane. Then the Schwarz representation formula is constructed by the C-auchy–Pompeiu formula and an explicit solution for the Schwarz boundary value problem for the inhomogeneous Cauchy–Riemann equation on the domain is presented. We also discuss about the condition of solvability and by using the Schwarz boundary value problem, the homogeneous Ne-umann and the inhomogeneous Dirichlet boundary value problems are investigated.
Journal Article
High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems
High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation.
Journal Article
Eta-Invariant of Elliptic Parameter-Dependent Boundary-Value Problems
In this paper, we study the eta-invariant of elliptic parameter-dependent boundary-value problems and its main properties. Using Melrose’s approach, we define the eta-invariant as a regularization of the winding number of the family. In this case, the regularization of the trace requires obtaining the asymptotics of the trace of compositions of invertible parameter-dependent boundaryvalue problems for large values of the parameter. Obtaining the asymptotics uses the apparatus of pseudodifferential boundary-value problems and is based on the reduction of parameter-dependent boundary-value problems to boundary-value problems with no parameter.
Journal Article
AN ENERGY STABLE AND CONVERGENT FINITE-DIFFERENCE SCHEME FOR THE MODIFIED PHASE FIELD CRYSTAL EQUATION
We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step-size. We present a local-in-time error estimate that ensures the pointwise convergence of the scheme.
Journal Article
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the \"probability error\" with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.
Journal Article
Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity
The classical Łojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, and tame geometry. This paper provides alternative characterizations of this type of inequality for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In the framework of metric spaces, we show that a generalized form of the Łojasiewicz inequality (hereby called the Kurdyka-Łojasiewicz inequality) is related to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by −∂f-\\partial f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka-Łojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines —a concept linked to the location of the less steepest points at the level sets of ff— and integrability conditions are given. In the convex case these results are significantly reinforced, allowing us in particular to establish a kind of asymptotic equivalence for discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2C^{2} function in R2\\mathbb {R}^{2} is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka-Łojasiewicz inequality.
Journal Article