Purpose

The purpose of this study is to extend a novel numerical method proposed by the first author, known as the dual mesh control domain method (DMCDM), for the solution of linear differential equations to the solution of nonlinear heat transfer and like problems in one and two dimensions.

Design/methodology/approach

In the DMCDM, a mesh of finite elements is used for the approximation of the variables and another mesh of control domains for the satisfaction of the governing equation. Both meshes fully cover the domain but the nodes of the finite element mesh are inside the mesh of control domains. The salient feature of the DMCDM is that the concept of duality (i.e. cause and effect) is used to impose boundary conditions. The method possesses some desirable attributes of the finite element method (FEM) and the finite volume method (FVM).

Findings

Numerical results show that he DMCDM is more accurate than the FVM for the same meshes used. Also, the DMCDM does not require the use of any ad hoc approaches that are routinely used in the FVM.

Originality/value

To the best of the authors’ knowledge, the idea presented in this work is original and novel that exploits the best features of the best competing methods (FEM and FVM). The concept of duality is used to apply gradient and mixed boundary conditions that FVM and its variant do not.

中文翻译：

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求解非线性方程组的新数值方法及其在传热中的应用

目的

这项研究的目的是将第一作者提出的一种新颖的数值方法（称为双重网格控制域方法（DMCDM））扩展到线性微分方程的解到非线性传热等问题的求解。二维。

设计/方法/方法

在DMCDM中，有限元网格用于变量的逼近，而控制域的另一个网格用于满足控制方程。两个网格都完全覆盖了该域，但是有限元网格的节点都在控制域的网格内。DMCDM的显着特征是使用对偶（即因果）概念来施加边界条件。该方法具有有限元方法（FEM）和有限体积方法（FVM）的一些理想属性。

发现

数值结果表明，对于所使用的相同网格，DMCDM比FVM更为准确。同样，DMCDM不需要使用FVM中常规使用的任何临时方法。

创意/价值

就作者所知，本文提出的想法是新颖新颖的，它利用了最佳竞争方法（FEM和FVM）的最佳功能。对偶性的概念用于应用FVM及其变体不提供的梯度和混合边界条件。