The Galerkin method is often employed for numerical integration of evolutionary equations, such as the Navier-Stokes equation or the magnetic induction equation. Application of the method requires solving an equation of the form $P(Av-f)=0$ at each time step, where $v$ is an element of a finite-dimensional space $V$ with a basis satisfying boundary conditions, $P$ is the orthogonal projection on this space and $A$ is a linear operator. Usually the coefficients of $v$ expanded in the basis are found by calculating the matrix of $PA$ acting on $V$ and solving the respective system of linear equations. For physically realistic boundary conditions (such as the no-slip boundary conditions for the velocity, or for a dielectric outside the fluid volume for the magnetic field) the basis is often not orthogonal and solving the problem can be computationally demanding. We propose an algorithm giving an opportunity to reduce the computational cost for such a problem. Suppose there exists a space $W$ that contains $V$, the difference between the dimensions of $W$ and $V$ is small relative to the dimension of $V$, and solving the problem $P(Aw-f)=0$, where $w$ is an element of $W$, requires less operations than solving the original problem. The equation $P(Av-f)=0$ is then solved in two steps: we solve the problem $P(Aw-f)=0$ in $W$, find a correction $h=v-w$ that belongs to a complement to $V$ in $W$, and obtain the solution $w+h$. When the dimension of the complement is small the proposed algorithm is more efficient than the traditional one.

翻译：伽辽金方法常用于演化方程的数值积分，例如纳维-斯托克斯方程或磁感应方程。应用该方法需要每时间步求解形如 $P(Av-f)=0$ 的方程，其中 $v$ 是有限维空间 $V$ 的元素，该空间具有满足边界条件的基函数，$P$ 是该空间上的正交投影算子，$A$ 为线性算子。通常通过计算 $PA$ 在 $V$ 上的作用矩阵并求解相应线性方程组来获得 $v$ 在基函数展开下的系数。对于物理真实边界条件（如速度的无滑移边界条件，或磁场在流体体积外为电介质的条件），基函数通常不具正交性，导致求解问题计算量极大。我们提出一种算法，可在此类问题中降低计算成本。假设存在空间 $W$ 包含 $V$，且 $W$ 与 $V$ 的维度差相对于 $V$ 的维度较小，同时求解 $P(Aw-f)=0$（其中 $w$ 为 $W$ 的元素）所需的运算量少于原问题。则方程 $P(Av-f)=0$ 可分两步求解：首先在 $W$ 中求解 $P(Aw-f)=0$，再找出属于 $W$ 中 $V$ 的补空间的修正项 $h=v-w$，最终得到解 $w+h$。当补空间维度较小时，所提算法比传统方法更高效。