Local Plus Global Adaptivity in Moving Node Finite Element Methods

9706353 Keith Miller Miller计划研究移动节点有限元方法的局部加全局适应性。他计划首先（与普渡大学的尼尔·卡尔森合作）将二维梯度加权移动有限元的局部适应性（通过其节点的连续运动）与改进的全局适应性（通过基于丰富的几何标准和局部误差估计的节点的添加、删除和重新连接）结合起来。在这里，他们在开发GWMFE方法方面建立了长期而富有成效的合作。这种方法特别适合于发展出尖锐的运动锋面的问题，特别是需要解决锋面的精细结构的问题。GWMFE在局部移动其节点以解决解决方案的尖锐特征方面做得非常好，但如果我们要拥有真正健壮和灵活的代码，显然需要全局适应性。在这里，他们将对Miller的学生Kuprat在其1992年的论文中引入的部分全局适应性进行根本性的改变和补充，这极大地增加了该方法的稳健性。米勒的第二项计划是开发一个带有流线扩散的修正GWMFE；这将纠正当前方法在处理某些稳态对流问题时存在的问题，即GWMFE网格随流向下游漂移而不是达到稳态。这个修改后的GWMFE在1-D中工作得非常好；成功地将其扩展到多维将是慢瞬态和近稳态流体计算的重大进步。二维三角形网格的有限元方法通常计算一个偏微分方程（PDE）或一组偏微分方程的分段线性近似解；也就是说，近似解的图是一个具有平面三角形面的演化曲面。对于标准有限元方法，网格是指定的和固定的；然而，对于GWMFE方法，网格是允许变形的，网格的节点自己决定如何移动。在锋面急剧移动的问题上，节点可以自动集中在锋面上并随之移动。与标准方法相比，使用更少的节点和更大的时间步长，就可以在解的关键区域获得高分辨率。例如，硅芯片制造中掺杂砷离子的高度非线性扩散，半导体器件切换状态时空穴、电子和电压纳秒演化的漂移扩散方程，以及油藏驱油的“黑油”方程。GWMFE方法在这些问题和许多其他问题上取得了惊人的成功，甚至在逻辑固定网格的约束下也是如此；也就是说，网格自动变形，但其节点的数量和相互连接保持固定。然而，逻辑固定的网格对于许多重要问题来说是不够的，很明显，通过增加全局适应性（根据需要插入和删除节点）可以大大提高方法的效率和鲁棒性。这需要大量的代码开发，但是GWMFE中局部和全局自适应的结合应该会产生一个精确和鲁棒的组合方法，它使用更少的节点，从而可以有效地计算具有移动尖锐特征的重要类别的问题，而这些问题目前是无法通过数值计算实现的。

9706353 Keith Miller Miller plans to work on local plus global adaptivity for moving node finite element methods. He plans first (in collaboration with Neil Carlson of Purdue) to combine the local adaptivity of 2-D gradient-weighted Moving Finite Elements (through the continuous movement of its nodes) with an improved global adaptivity (through addition, deletion and reconnection of nodes, based upon enriched geometric criteria and upon local error estimates). Here they are building upon their longstanding and fruitful collaboration in developing the GWMFE method. This method is especially suited to problems which develop sharp moving fronts, especially problems where one needs to resolve the fine-scale structure of the fronts. GWMFE does an extremely fine job of moving its nodes around locally to resolve the sharp features of the solution, but clearly global adaptivity is needed if we are to have truly robust and flexible codes. Here they would be making fundamental changes and additions to the partial global adaptivity introduced by Miller's student Kuprat in his thesis of '92, which added so greatly to the robustness of the method. Miller plans second to develop a revised GWMFE with streamline diffusion; this should correct some problems which the present method has on certain steady-state convection problems in which the GWMFE grid drifts downstream with the flow rather than coming to a steady-state. This revised GWMFE works extremely well in 1-D; success in extending it to multidimensions would be a significant advance for slow-transient and near steady-state fluid computations. Finite element (FE) methods with a triangular grid in 2-D typically compute a piecewise linear approximation to the solution of a partial differential equation (PDE) or system of PDEs; that is, the graph of the approximate solution is an evolving surface with planar triangular faces. For standard FE methods the grid is specified and fixed; however, for the GWMFE method the grid is allowed to deform and the nodes of the grid decide for themselves how to move. On problems with sharp moving fronts the nodes can thus automatically concentrate in the front and move with it. In this way one attains high resolution in the critical regions of the solution while using far fewer nodes and far larger time steps than with standard methods. Examples are the highly nonlinear diffusion of doped arsenic ions in the manufacture of silicon chips, the drift-diffusion equations for the nanosecond evolution of holes, electrons and voltages as a semiconductor device switches states, and the "black oil" equations for flooding of oil reservoirs. The GWMFE method has been strikingly successful on these and many other problems even within the constraints of a logically-fixed grid; that is, the grid deforms automatically, but the number and interconnections of its nodes remain fixed. However, a logically-fixed grid is inadequate for many important problems and it is apparent that the efficiency and robustness of the method can be greatly enhanced by adding global adaptivity (the insertion and deletion of nodes as needed). This requires a good deal of code development, but the combination of local plus global adaptivity in GWMFE should yield a combined method which is accurate and robust, which uses far fewer nodes, and which thereby renders efficiently computable important classes of problems with moving sharp features which are presently inaccessible to numerical computation.