Efficient Spectral-Galerkin Method and Analysis for Elliptic PDEs with Non-local Boundary Conditions

Abstract

We present an efficient Legendre–Galerkin method and its error analysis for a class of PDEs with non-local boundary conditions. We also present several numerical experiments, including the scattering problem from an open cavity, to demonstrate the accuracy and efficiency of the proposed method.

Appendix: Matrix Diagonalization Method

In this section we briefly recall the matrix diagonalization method in [33] for solving the linear system $A\mathbf {u}=\mathbf {f}$ where A is the matrix defined in (3.9). We can rewrite it as the following matrix equation:

$$\begin{aligned} \alpha M_x U M_y^{T} + S_x U M_y^{T} + M_x U S_y^{T} = F. \end{aligned}$$

(6.1)

We diagonalize in x-direction and reduce the problem to $N+1$ one-dimension equations (in y-direction) following the steps below:

• 1. Consider the generalized eigenvalue problem:

  $$\begin{aligned} M_x \bar{x} = \lambda S_x \bar{x}. \end{aligned}$$

  (6.2)

  $M_x$ and $S_x$ are symmetric positive definite matrices. Let $\varLambda$ be the diagonal matrix whose diagonal entries $\lambda _p$ are the eigenvalues of (6.2), and let E be the matrix whose columns are the eigenvectors of (6.2). We have

  $$\begin{aligned} M_x E = S_x E \varLambda ,\quad E^{-1}=E^T. \end{aligned}$$

  (6.3)

• 2. Let $U=EV$, thanks to (6.3), the equation (6.1) becomes

  $$\begin{aligned} \alpha S_x E \varLambda V M_y^{T} + S_x EV M_y^{T} + S_x E \varLambda V S_y^{T} = F. \end{aligned}$$

  Multiplying $E^{T}S_x^{-1}$ to both sides of the above equation yields

  $$\begin{aligned} \alpha \varLambda V M_y^{T} + V M_y^{T} + \varLambda V S_y^{T} = E^{T}S_x^{-1}F:=G. \end{aligned}$$

  (6.4)

• 3. Let $\mathbf {v}_p = (v_{p0},v_{p1},\ldots ,v_{p\scriptscriptstyle {N}})^{T}$ and $\mathbf {g}_p = (g_{p0},g_{p1},\ldots , g_{p\scriptscriptstyle {N}})^{T}, 0\le p\le N$. Then the p-th row of the equation (6.4) can be written as

  $$\begin{aligned} ((\alpha \lambda _p +1) M_y + \lambda _p S_y) \mathbf {v}_p = \mathbf {g}_p. \end{aligned}$$

  (6.5)

Since $M_y$ and $S_y$ are sparse, we can solve (6.5) in O(N) operations for each p. Hence, the main cost of solving (6.1) is the two matrix-matrix multiplications which cost a small multiple of $N^3$ operations.