Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes logtj=loga+logtNajNr,j=0,1,2,⋯,N with a≥1 and r≥1, where loga=logt0<logt1<⋯<logtN=logT is a partition of [logt0,logT]. We also consider the rectangular and trapezoidal methods for solving Caputo–Hadamard fractional differential equations with the non-uniform meshes logtj=loga+logtNaj(j+1)N(N+1),j=0,1,2,⋯,N. Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., DCHa,tαy(t)∉C1[a,T] with α∈(0,2), the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio r≥1. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

We propose a p-multilevel preconditioner for hybrid high-order (HHO) discretizations of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical discontinuous Galerkin scheme. An efficient implementation is proposed where coarse level operators are inherited using $$L^2$$ L 2 -orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes. For the two HHO formulations, featuring discontinuous or hybrid pressure, we study how the combination of p-coarsening and static condensation influences the V-cycle iteration. In particular, two different static condensation procedures are considered for the discontinuous pressure HHO variant, resulting in global linear systems with a different number of unknowns and matrix non-zero entries. Interestingly, we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.

Approximating inverse FEM matrices on non-uniform meshes with $${\mathcal{H}}$$-matrices

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $${\mathcal{H}}$$ H -matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the $${\mathcal{H}}$$ H -matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.