[Paper Review] Space-time Galerkin isogeometric method and efficient solver for parabolic problems

Gabriele Loli, Monica Montardini+2 more|arXiv (Cornell University)|Sep 16, 2019

Advanced Numerical Analysis Techniques4 citations

TL;DR

This paper proposes a space-time Galerkin isogeometric method with a novel preconditioner based on Kronecker product structures and an extended Fast Diagonalization method, enabling efficient solution of parabolic problems. The approach ensures near-linear scalability with degrees of freedom and maintains robustness across polynomial degrees and complex geometries with variable coefficients.

ABSTRACT

In this work we focus on the preconditioning of a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, we propose a preconditioner that is the sum of Kronecker products of matrices and that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. polynomial degree and the time required for the application is almost proportional to the number of degrees-of-freedom, for a serial execution. By incorporating some information on the geometry parametrization and on the equation coefficients, we keep high efficiency with non-trivial domains and variable thermal conductivity and heat capacity coefficients.

Motivation & Objective

• Address the computational inefficiency of solving parabolic problems via space-time isogeometric discretizations.
• Develop a preconditioner that maintains efficiency across varying polynomial degrees and complex geometries.
• Incorporate geometric parametrization and variable coefficients (e.g., thermal conductivity) into the solver without sacrificing performance.
• Achieve near-linear time complexity per solve for serial execution, ensuring scalability.

Proposed method

• Utilize a Galerkin space-time isogeometric discretization of the heat equation with tensor-product basis functions.
• Construct a preconditioner as a sum of Kronecker products of spatial and temporal matrices to exploit tensor structure.
• Extend the classical Fast Diagonalization method to efficiently apply the preconditioner in nearly optimal time.
• Integrate geometric parametrization and variable coefficients (e.g., thermal conductivity and heat capacity) directly into the preconditioner design.
• Ensure the preconditioner remains effective and efficient even for non-trivial domains and non-constant coefficients.

Experimental results

Research questions

• RQ1 Can a space-time isogeometric method achieve near-linear time complexity in solving parabolic problems using a preconditioner based on Kronecker structures?
• RQ2 How does the proposed preconditioner perform across varying polynomial degrees in the isogeometric discretization?
• RQ3 To what extent can the solver maintain efficiency when handling complex geometries and variable coefficients?
• RQ4 Can the extended Fast Diagonalization method efficiently apply the preconditioner in a serial execution setting?

Key findings

• The proposed preconditioner enables the solution time to scale almost linearly with the number of degrees of freedom in serial execution.
• The method remains robust with respect to the polynomial degree of the isogeometric basis functions.
• The inclusion of geometric parametrization and variable coefficients in the preconditioner does not degrade solver efficiency.
• The extended Fast Diagonalization method effectively applies the preconditioner by leveraging the tensor product structure of the basis functions.