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

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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.

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Correspondence to Jie Shen.

Additional information

The work of J.S. is partially supported by NFSC grants 11371298 and 11421110001.

Appendix: Matrix Diagonalization 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. 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. 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. 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.

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Hu, L., Ma, L. & Shen, J. Efficient Spectral-Galerkin Method and Analysis for Elliptic PDEs with Non-local Boundary Conditions. J Sci Comput 68, 417–437 (2016). https://doi.org/10.1007/s10915-015-0145-x

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  • DOI: https://doi.org/10.1007/s10915-015-0145-x

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