Approximating inverse FEM matrices on non-uniform meshes with -matrices

Area of Science:

• Numerical Analysis
• Computational Mechanics
• Scientific Computing

Background:

• Finite element analysis (FEA) often involves large, dense stiffness matrices.
• Direct solvers for FEA are computationally expensive due to matrix size and density.
• Hierarchical matrices (H-matrices) offer a data-sparse representation for matrices.

Purpose of the Study:

• To investigate the approximation of the inverse of the finite element stiffness matrix in the H-matrix format.
• To establish the efficiency and accuracy of H-matrix approximation for stiffness matrices.
• To explore the utility of H-matrix approximations as solvers or preconditioners.

Main Methods:

• Consideration of shape-regular, potentially non-uniform meshes, including algebraically graded meshes.
• Mathematical proof of approximation rates for the stiffness matrix inverse in H-matrix format.
• Analysis of storage complexity and computational efficiency of H-matrix representations.

Main Results:

• The inverse of the stiffness matrix can be approximated in the H-matrix format at an exponential rate concerning the block rank.
• H-matrix storage complexity is logarithmic-linear and grows linearly with block rank.
• Efficient approximations are achievable for a broad class of meshes.

Conclusions:

• H-matrix approximation provides an efficient method for representing the inverse of finite element stiffness matrices.
• These approximations are suitable for use as approximate direct solvers or preconditioners in iterative solvers.
• The findings enable significant computational savings in FEA.