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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

This work aims to develop a fast and spatially fourth-order Cartesian grid finite difference method for solving elliptic and parabolic problems over two-dimensional irregular domains with sharply curved boundaries, under the assumptions that the boundary is C 1 -continuous and the solution is sufficiently smooth up to the boundary. The proposed Augmented Matched Interface and Boundary (AMIB) method inherits its predecessor’s speed and accuracy advantages, such as maintaining the Fast Fourier-Transform (FFT) efficiency and being fourth-order accuracy in handling any boundary conditions (Dirichlet, Neumann, Robin, or their combinations). To accommodate sharply curved boundaries, the proposed adaptive AMIB method features two significant improvements. First, an adaptive ray-casting Matched Interface and Boundary (MIB) scheme was developed to overcome the difficulties of generating fictitious values at some grid points where the boundary is sharply curved by reusing previously calculated fictitious values at nearby grid points. Second, several stabilizers, which include a new preconditioner for the resulting augmented system and proper grid selection requirements to interpolate the fictitious values and approximate derivative jumps in the MIB scheme, were designed to ensure the stability of the AMIB method. Numerical experiments have been conducted to validate the proposed AMIB method for solving boundary and initial value problems with sharply curved boundaries.

The first boundary value problem for the heat equation in a cone is considered in the case of a domain degenerating at the initial time. The eigenfunctions of the problem are found. Estimates of the Green’s function are obtained. For the problem with a zero boundary function, unique solvability in some class of functions admitting certain growth as they approach the vertex of the cone is established. Similar results are obtained for a cone degenerating at the final moment of time. Additionally, the first boundary value problem in domains degenerating only with respect to some of the variables is con-sidered.

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.

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.

(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.

The equations of the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows are considered in a bounded domain. The viscosity coefficients and heat conductivity can depend on the temperature. A solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global variational weak solution to the three-dimensional full magnetohydrodynamic equations with large data is established.

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.

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.

In this work, we establish Lyapunov-type inequalities for the fractional boundary value problems with Caputo–Hadamard fractional derivative subject to multipoint and integral boundary conditions. As far as we know, there is no literature that has studied these problems.