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compute_FSBP.m
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compute_FSBP.m
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%% compute_FSBP
%
% Description:
% Function to compute FSBP operators
%
% Author: Jan Glaubitz
% Date: Jan 07, 2022
%
% INPUT:
% basis_F : basis of the approximation space F
% dx_basis_F : derivatives of the basis elements
% x : grid points
% w : weights of a positive and (F^2)'-exact QF
%
% OUTPUT:
% D : differentiation matrix
% P : diagonal-norm matrix
% Q : matrix for boundary correction
function [D, P, Q] = compute_FSBP( basis_F, dx_basis_F, x, w )
N = length(x); % number of grid points
K = length( basis_F(x(1)) ); % dimension of F
%% Diagonal-norm matrix P
P = sparse(diag(w));
%% Anti-symmetric part of Q, Q_anti
% Prepare matrices
F = zeros(N,K); % Vandermonde-like matrix
F_x = zeros(N,K); % Vandermonde-like matrix for the derivatives
for n=1:N
F(n,:) = basis_F( x(n) )';
F_x(n,:) = dx_basis_F( x(n) )';
end
B = zeros(N); B(1,1) = -1; B(end,end) = 1; % boundary matrix
R = P*F_x - 0.5*B*F; % right-hand side of the matrix equation for Q_anti
% Vectorize
A = kron( F', eye(N) ); % coefficient matrix for vectorized version
r = R(:); % vectorized right-hand side
% Commutation matrix C
I = reshape(1:N*N, [N, N]); % initialize a matrix of indices
I = I'; % transpose it
I = I(:); % vectorize the required indices
C = speye(N*N); % Initialize an identity matrix
C = C(I,:); % Re-arrange the rows of the identity matrix
% Get the anti-symmetric least-squares solution to
A_ext = [ A; C+speye(N^2) ];
r_ext = [ r; zeros(N^2,1) ];
q_anti = lsqminnorm(A_ext,r_ext); % least-squares solution
Q_anti = reshape(q_anti,N,N);
%% Q and the differentiation matrix D
Q = Q_anti + 0.5*B; % matrix Q
P_inv = diag(1./w); % inverse of P
D = P_inv*Q; % differentiation matrix D
end