The normal derivative lemma for the Laplacian on a polyhedral set

2014 ◽  
Vol 96 (1-2) ◽  
pp. 122-129
Author(s):  
S. N. Oshchepkova ◽  
O. M. Penkin ◽  
D. V. Savasteev
Keyword(s):  
Normal Derivative ◽  
Polyhedral Set

A High-Order Flux Reconstruction Method for 2-D Vorticity Transport

2021 ◽  
Author(s):  
Adrin Gharakhani
Keyword(s):  
Finite Difference ◽  
Poisson Equation ◽  
Finite Difference Methods ◽  
Neumann Boundary ◽  
Normal Derivative ◽  
High Order ◽  
Benchmark Problems ◽  
Flux Reconstruction ◽  
Reconstruction Method ◽  
Vorticity Transport

Abstract A compact high-order finite difference method on unstructured meshes is developed for discretization of the unsteady vorticity transport equations (VTE) for 2-D incompressible flow. The algorithm is based on the Flux Reconstruction Method of Huynh [1, 2], extended to evaluate a Poisson equation for the streamfunction to enforce the kinematic relationship between the velocity and vorticity fields while satisfying the continuity equation. Unlike other finite difference methods for the VTE, where the wall vorticity is approximated by finite differencing the second wall-normal derivative of the streamfunction, the new method applies a Neumann boundary condition for the diffusion of vorticity such that it cancels the slip velocity resulting from the solution of the Poisson equation for the streamfunction. This yields a wall vorticity with order of accuracy consistent with that of the overall solution. In this paper, the high-order VTE solver is formulated and results presented to demonstrate the accuracy and convergence rate of the Poisson solution, as well as the VTE solver using benchmark problems of 2-D flow in lid-driven cavity and backward facing step channel at various Reynolds numbers.

scholarly journals Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Ihor Borachok ◽  
Roman Chapko ◽  
B. Tomas Johansson
Keyword(s):  
Cauchy Problem ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Single Layer ◽  
Normal Derivative ◽  
Boundary Integral ◽  
Iteration Step ◽  
3 Dimensional ◽  
Boundary Surfaces

AbstractWe consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert’s method [

Cylindrical optimal rearrangement problem leading to a new type obstacle problem

2018 ◽  
Vol 24 (2) ◽  
pp. 859-872 ◽  
Author(s):  
Hayk Mikayelyan
Keyword(s):  
Comparison Principle ◽  
Obstacle Problem ◽  
Normal Derivative ◽  
Force Function ◽  
Cylindrical Domain ◽  
New Type ◽  
Existence And Regularity Results ◽  
Regularity Results ◽  
Cylindrical Axis

An optimal rearrangement problem in a cylindrical domainΩ=D× (0, 1) is considered, under the constraint that the force function does not depend on thexnvariable of the cylindrical axis. This leads to a new type of obstacle problem in the cylindrical domain     Δu(x′,xn) =χ{v>0}(x′) +χ{v=0}(x′) [∂νu(x′,0) +∂νu(x′, 1)]arising from minimization of the functional     ∫Ω½;|∇u(x)|2+χ{v>0}(x′)u(x) dx,wherev(x′) =∫01u(x′,t)dt, and∂νuis the exterior normal derivative ofuat the boundary. Several existence and regularity results are proven and it is shown that the comparison principle does not hold for minimizers.

scholarly journals Normal derivative for bounded domains with general boundary

1988 ◽  
Vol 308 (2) ◽  
pp. 785-785 ◽  
Author(s):  
Guang Lu Gong ◽  
Min Ping Qian ◽  
Martin L. Silverstein
Keyword(s):  
Normal Derivative ◽  
General Boundary ◽  
Bounded Domains

Jacobi Matrices and the Spectrum of the Neumann Operator on a Family of Riemann Surfaces

1993 ◽  
Vol 45 (4) ◽  
pp. 709-726
Author(s):  
Julian Edward
Keyword(s):  
Harmonic Function ◽  
Riemann Surfaces ◽  
Smooth Manifold ◽  
Hypergeometric Functions ◽  
Normal Derivative ◽  
Jacobi Matrices ◽  
Uniform Bounds ◽  
Neumann Operator ◽  
Direct Sum ◽  
Asymptotics Of The Eigenvalues

AbstractThe Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. The spectrum of the Neumann operator is studied on the curves bounding a family of Riemann surfaces. The Neumann operator is shown to be isospectral to a direct sum of symmetric Jacobi matrices, each acting on l2(ℤ). The Jacobi matrices are shown to be isospectral to generators of bilateral, linear birth-death processes. Using the connection between Jacobi matrices and continued fractions, it is shown that the eigenvalues of the Neumann operator must solve a certain equation involving hypergeometric functions. Study of the equation yields uniform bounds on the eigenvalues and also the asymptotics of the eigenvalues as the curves degenerate into a wedge of circles.

scholarly journals A STRONGLY ILL-POSED INTEGRO-DIFFERENTIAL PARABOLIC PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS

2013 ◽  
Vol 18 (3) ◽  
pp. 395-414
Author(s):  
Alfredo Lorenzi
Keyword(s):  
Parabolic Problem ◽  
Dirichlet Boundary ◽  
Integral Boundary Conditions ◽  
Integral Condition ◽  
Normal Derivative ◽  
Integral Boundary ◽  
Additional Information ◽  
Ill Posed ◽  
Linear Integral ◽  
Continuous Dependence Results

Via Carleman estimates we prove uniqueness and continuous dependence results for an identification and strongly ill-posed linear integro-differential parabolic problem with the Dirichlet boundary condition, but with no initial condition. The additional information consists in a boundary linear integral condition involving the normal derivative of the temperature on the whole of the lateral boundary of the space-time domain.

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