Least Squares PGD Mimetic Spectral Element Methods for Systems of First-Order PDEs

许多描述科学和工程问题的数学模型都是在高维空间中定义的。例子可以在各种应用领域找到，如量子化学，材料（包括复杂流体）的动力学理论描述，控制许多生物过程的化学主方程（例如细胞信号）和金融数学模型（例如期权定价）。这些问题的数学描述总是以偏微分方程（PDE）系统的形式出现。对于实际问题，这些系统不具有解析解，因此有必要对其进行数值求解。重要的是，作为离散化的结果获得的代数方程组的系统是一个兼容的（模仿）和物理上一致的系统，使数值近似是一个准确的表示的物理解决方案的问题。偏微分方程的控制系统是写在一个等价的一阶微分方程系统，随后制定的最小二乘泛函方面。有效地将偏微分方程系统的解转化为无约束最小化问题。最小二乘公式的特殊稳定性导致了低阶有限元在其离散化中的广泛使用。不幸的是，这些方法只是近似保守的，这通常会导致违反基本的物理性质，如质量守恒的损失。在许多情况下，这个缺点可能超过最小二乘法的潜在优势。因此，提高最小二乘方法的守恒性是至关重要的。

Many mathematical models that describe problems in science and engineering are defined in high dimensional spaces. Examples can be found in a diverse range of application areas such as quantum chemistry, kinetic theory descriptions of materials (including complex fluids), the chemical master equation governing many biological processes (e.g. cell signalling) and models of financial mathematics (e.g. option pricing). The mathematical description of these problems is invariably in terms of a system of partial differential equations (PDEs). For practical problems these systems do not possess analytical solutions and therefore it is necessary to solve them numerically. It is important that the system of algebraic equations obtained as a result of discretisation is a compatible (mimetic) and physically consistent system so that the numerical approximation is an accurate representation of the physical solution to the problem. The governing systems of PDEs is written in terms of an equivalent system of first-order differential equations which is subsequently formulated in terms of a least squares functional. Effectively the solution of a system of PDEs is converted into an unconstrained minimisation problem. The exceptional stability of least-squares formulations has led to the widespread use of low-order finite elements in their discretization. Unfortunately, these methods are only approximately conservative, which generally leads to violation of fundamental physical properties, such as loss of mass conservation. In many cases this drawback can outweigh the potential advantages of least squares methods. As a result, improving the conservation properties of least-squares methods is crucially important.