Efficient CUDA Polynomial Preconditioned Conjugate Gradient Solver for Finite Element Computation of Elasticity Problems

Algorithm 6

Mixed precision polynomial PCG.

𝐀^𝐿=𝐀^𝐻

𝐫^𝐻₀=𝐛^𝐻−𝐀^𝐻𝐱^𝐻₀

𝐫^𝐿₀=𝐫^𝐻₀

𝐳^𝐿₀=𝑃_𝑚(𝐀^𝐿)𝐫^𝐿₀

𝐳^𝐻₀=𝐳^𝐿₀

𝐩^𝐻₀=𝐳^𝐻₀

for𝑗=0,1,…, until  convergence

 𝛼^𝐻_𝑗=(𝐫^𝐻_𝑗,𝐳^𝐻_𝑗)/(𝐀^𝐻𝐩^𝐻_𝑗,𝐩^𝐻_𝑗)

 𝐱^𝐻_𝑗+1=𝐱^𝐻_𝑗+𝛼^𝐻_𝑗𝐩^𝐻_𝑗

 𝐫^𝐻_𝑗+1=𝐫^𝐻_𝑗−𝛼_𝑗𝐀^𝐻𝐩^𝐻_𝑗

 𝐫^𝐿_𝑗+1=𝐫^𝐻_𝑗+1

 𝐳^𝐿_𝑗+1=𝑃_𝑚(𝐀^𝐿)𝐫^𝐿_𝑗+1

 𝐳^𝐻_𝑗+1=𝐳^𝐿_𝑗+1

 𝛽^𝐻_𝑗=(𝐫^𝐻_𝑗+1,𝐳^𝐻_𝑗+1)/(𝐫^𝐻_𝑗,𝐳^𝐻_𝑗)

 𝐩^𝐻_𝑗+1=𝐳^𝐻_𝑗+1+𝛽^𝐻_𝑗𝐩^𝐻_𝑗

end  for