Fast Domain Decomposition Type Solver for Stiffness Matrices of Reference p-Elements

2013 ◽  
Vol 13 (2) ◽  
pp. 161-183 ◽  
Author(s):  
Vadim Korneev
Keyword(s):  
Finite Difference ◽  
Domain Decomposition ◽  
Elliptic Equations ◽  
Harmonic Functions ◽  
Schur Complement ◽  
P Elements ◽  
Stiffness Matrices ◽  
The Family ◽  
Discrete Harmonic Functions

Abstract. A key component of domain decomposition solvers for hp discretizations of elliptic equations is the solver for internal stiffness matrices of p-elements. We consider an algorithm which belongs to the family of secondary domain decomposition solvers, based on the finite-difference like preconditioning of p-elements, and was outlined by the author earlier. We remove the uncertainty in the choice of the coarse (decomposition) grid solver and suggest the new interface Schur complement preconditioner. The latter essentially uses the boundary norm for discrete harmonic functions induced by orthotropic discretizations on slim rectangles, which was derived recently. We prove that the algorithm has linear arithmetical complexity.

scholarly journals Discrete harmonic functions on an orthant in $\mathbb{Z}^d$

2015 ◽  
Vol 20 (0) ◽  
Author(s):  
Mustapha Sami ◽  
Aymen Bouaziz ◽  
Mohamed Sifi
Keyword(s):  
Harmonic Functions ◽  
Discrete Harmonic Functions

Convergence and applications of reproducing kernels for classes of discrete harmonic functions

1974 ◽  
Vol 11 (3) ◽  
pp. 339-358
Author(s):  
C. Wayne Mastin
Keyword(s):  
Harmonic Functions ◽  
Reproducing Kernels ◽  
Biharmonic Operator ◽  
Convergence Properties ◽  
Interpolation Problems ◽  
Discrete Analogs ◽  
Discrete Harmonic Functions

This paper gives convergence properties and applications of the discrete analogs of reproducing kernels for various families of harmonic functions. From these results information is obtained on the solution of interpolation problems, the convergence of the discrete Neumann's function, and the solution to problems involving the discrete biharmonic operator.

scholarly journals Stability estimates for discrete harmonic functions on product domains

2013 ◽  
Vol 7 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Maru Guadie
Keyword(s):  
Fourier Series ◽  
Dirichlet Problem ◽  
Harmonic Measure ◽  
Harmonic Functions ◽  
Discrete Analog ◽  
Stability Estimates ◽  
Product Domains ◽  
Discrete Harmonic Functions

We study the Dirichlet problem for discrete harmonic functions in unbounded product domains on multidimensional lattices. First we prove some versions of the Phragm?n-Lindel?f theorem and use Fourier series to obtain a discrete analog of the three-line theorem for the gradients of harmonic functions in a strip. Then we derive estimates for the discrete harmonic measure and use elementary spectral inequalities to obtain stability estimates for Dirichlet problem in cylinder domains.

On discrete harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 194-206 ◽  
Author(s):  
H. A. Heilbronn
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Difference Equation ◽  
Harmonic Functions ◽  
The Difference ◽  
Discrete Harmonic Functions ◽  
Image Position

A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the difference equationThis equation can be considered as the direct analogue either of the differential equationor of the integral equationin the notation normally employed to harmonic functions.

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