A mimetic iterative scheme for solving biharmonic equations

A New Analysis of an Implicit Mimetic Scheme for the Heat Equation

“…In this article, a second order mimetic operators described in [8] [9] will be analyzed in the context of the transient diffusivity equation. This version of the mimetic operators has been successfully applied to elliptic equations [9] [10], transient diffusivity equations [11]- [19], reservoir flow problems [20] [21], the acoustic wave equation [22] [23], the biharmonic equation [24] and the biharmonic wave equation [25]. From all these references, the more relevant to this article are [11]- [18] [20] [21], which presents mimetic schemes for the heat or diffusivity equations, and a brief review of their content will be described.…”

Shape reconstruction for an inverse biharmonic problem from partial Cauchy data

“…We denote by $$$ n $$$ the outward unit normal to $$$ \mathrm{\partial \Omega } $$$, and $$$ u $$$ is a solution of the following boundary value problem: $$$$ {\Delta}^2u=\sum \limits_{i=1}^2\frac{\partial^4u}{\partial {x}_i^4}+2\sum \limits_{i<j}\frac{\partial^4u}{\partial {x}_i^2\partial {x}_j^2}=0\kern2em \mathrm{in}\kern0.4em \Omega, $$$$ with the Navier boundary condition [10] $$$$ \left\{\begin{array}{ll}u& ={u}_0,\kern2em \mathrm{on}\kern.5em {\Gamma}_m,\\ {}\Delta u& ={u}_2,\kern2em \mathrm{on}\kern.5em {\Gamma}_m,\end{array}\right. $$$$ and Robin conditions [8, 17] …”

An efficient multigrid solver for 3D biharmonic equation with a discretization by 25-point difference scheme

“…The biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory, phase-field models and Stokes flows. Due to the significance of the biharmonic equation, a large number of numerical methods for solving the biharmonic equations have been proposed [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Most of these works focus on two-dimensional case.…”