## What is/are Graded Meshes?

Graded Meshes - In this paper, we introduce a multi-domain Muntz-polynomial spectral collocation method with graded meshes for solving second kind Volterra integral equations with a weakly singular kernel.^{[1]}So, we use the L1 scheme on graded meshes for time discretization.

^{[2]}Among them, some popular methods were constructed by using the graded meshes due to the solution with low temporal regularity.

^{[3]}Moreover, for a short-memory process, the optimal strong convergence order of the method with distributions under uniform and graded meshes is discussed.

^{[4]}For solutions with singular behaviour near t = 0 caused by the weakly singular kernels, we prove optimal algebraic convergence rates for the h-version of the discontinuous Galerkin approximations on graded meshes.

^{[5]}For the weighted finite element method an absolute error value is by one or two orders of magnitude less than for the approximate generalized solution obtained by the FEM with graded meshes in the overwhelming majority of nodes.

^{[6]}Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes.

^{[7]}To overcome this difficulty, we use the L1 scheme on graded meshes in time, such that considered time steps are very small near the origin which compensate the singularity of the solution.

^{[8]}Iterative processes or corrected schemes become dispensable by the use of the Newton linearized method and graded meshes in the temporal direction.

^{[9]}We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes.

^{[10]}A second-order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel.

^{[11]}In this paper, we investigate the local superconvergence of the discontinuous Galerkin (DG) solutions on quasi-graded meshes for nonlinear delay differential equations with vanishing delay.

^{[12]}To deal with the weak singularity caused by the fractional derivative that the solution has in the initial layer, the well-known L1 scheme on graded meshes has been used for time discretization.

^{[13]}In this paper we investigate multigrid methods for a quad-curl problem on graded meshes.

^{[14]}

## time fractional derivative

The L 2 - 1 σ scheme is applied for the discretization of time fractional derivative on graded meshes and anisotropic finite element method (FEM) is employed for the spatial discretization.^{[1]}Therefore, in order to improve the convergence order, we discrete the Caputo time fractional derivative by a new \begin{document}$ L1-2 $\end{document} format on graded meshes, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes.

^{[2]}Firstly, the time fractional derivative term is discretized by classical L1 format on graded meshes, and the spatial derivative term is approximated by meshless method.

^{[3]}

## Special Graded Meshes

It is shown that optimal convergence rates can be obtained by using special graded meshes.^{[1]}The existence and uniqueness of collocation solutions are proved under two special graded meshes.

^{[2]}