A Twisted Block Tangential Filtering Decomposition Preconditioner

Algorithm 1

Solving 𝐌𝐱=𝐟.

(1) Solving (𝐸+𝑇)𝑦=𝑓

   𝑦₁=𝑇₁⁻¹𝑓₁,𝑦_𝑚=𝑇_𝑚⁻¹𝑓_𝑚.

   for 𝑖=2∶𝑗−1

     𝑦_𝑖=𝑇_𝑖⁻¹(𝑓_𝑖−𝐿_𝑖−1𝑦_𝑖−1),

   end.

   for 𝑖=𝑚−1∶−1∶𝑗+1

     𝑦_𝑖=𝑇_𝑖⁻¹(𝑓_𝑖−𝑈_𝑖𝑦_𝑖+1),

   end.

   𝑦_𝑗=𝑇_𝑗⁻¹(𝑓_𝑗−𝑈_𝑗𝑦_𝑗+1−𝐿_𝑗−1𝑦_𝑗−1).

(2) Solving (𝑇⁻¹𝐹+𝐼)𝑥=𝑦

   𝑥_𝑗=𝑦_𝑗,

   for 𝑖=𝑗−1∶−1∶1

     𝑥_𝑖=𝑦_𝑖−𝑇_𝑖⁻¹𝑈_𝑖𝑦_𝑖+1,

   end

   for 𝑖=𝑗+1∶𝑚

     𝑥_𝑖=𝑦_𝑖−𝑇_𝑖⁻¹𝐿_𝑖−1𝑦_𝑖−1,

   end.