In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with the tempered fractional derivative of order $α.$ Although some of its variants are considered in many recent numerical analysis works, there are still some significant differences. Here we first provide the regularity estimates of the solution. Then a modified $L1$ scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at $t → 0^+,$ while the five-point difference scheme is used in space. Stability and convergence are proved in the sense of $L^∞$ norm, getting a sharp error estimate $\mathscr{O}(\tau^{{\rm min}\{2−α,rα\}})$ on graded meshes. Furthermore, the constant multipliers in the analysis do not blow up as the order of Caputo fractional derivative $α$ approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence orders of the presented schemes.

In this paper, we design and analyze new residual-type a posteriori error estimators for the local discontinuous Galerkin (LDG) method applied to semilinear second-order elliptic problems in two dimensions of the type $−∆u = f(x, u).$ We use our recent superconvergence results derived in Commun. Appl. Math. Comput. (2021) to prove that the LDG solution is superconvergent with an order $p+2$ towards the $p$-degree right Radau interpolating polynomial of the exact solution, when tensor product polynomials of degree at most $p$ are considered as basis for the LDG method. Moreover, we show that the global discretization error can be decomposed into the sum of two errors. The first error can be expressed as a linear combination of two $(p+1)$-degree Radau polynomials in the $x-$ and $y−$ directions. The second error converges to zero with order $p+2$ in the $L^2$-norm. This new result allows us to construct a posteriori error estimators of residual type. We prove that the proposed a posteriori error estimators converge to the true errors in the $L^2$-norm under mesh refinement at the optimal rate. The order of convergence is proved to be $p+2.$ We further prove that our a posteriori error estimates yield upper and lower bounds for the actual error. Finally, a series of numerical examples are presented to validate the theoretical results and numerically demonstrate the convergence of the proposed a posteriori error estimators.

In this paper, we present first-order accurate numerical methods for solution of the heat equation with uncertain temperature-dependent thermal conductivity. Each algorithm yields a shared coefficient matrix for the ensemble set improving computational efficiency. Both mixed and Robin-type boundary conditions are treated. In contrast with alternative, related methodologies, stability and convergence are unconditional. In particular, we prove unconditional, energy stability and optimal-order error estimates. A battery of numerical tests are presented to illustrate both the theory and application of these algorithms.

We apply orthogonal spline collocation with splines of degree $r ≥ 3$ to solve, on the unit square, Poisson’s equation with Neumann boundary conditions. We show that the $H^1$ norm error is of order $r$ and explain how to compute efficiently the approximate solution using a matrix decomposition algorithm involving the solution of a symmetric generalized eigenvalue problem.

A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.

Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems with implicit or Robin-type interface jump conditions. We design new enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. A linear enriched method in 1D was recently developed using one enrichment function and we generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. The natural tensor product extension to the 2D case is demonstrated. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. We also establish superconvergence at all discretization nodes (including exact nodal values in special cases). Numerical examples are provided to confirm the theory. Finally, to prove the efficiency of the method for practical problems, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.