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This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear PDEs. The main idea of these algorithms is to utilize the solutions on the kth-level adaptive meshes to find the solutions on the (k + 1)th-level adaptive meshes which are constructed by performing adaptive element bisections on the k th-level adaptive meshes. These algorithms transform nonsymmetric positive definite (non-SPD) PDEs (resp., nonlinear PDEs) into symmetric positive definite (SPD) PDEs (resp., linear PDEs). The proposed algorithms are both accurate and efficient due to the following advantages: They do not need to solve the nonsymmetric or nonlinear systems; the degrees of freedom are very small, they are easily implemented, and the interpolation errors are very small. Next, this paper constructs residual-type a posteriori error estimators which are shown to be reliable and efficient. The key ingredient in proving the efficiency is to establish an upper bound of the oscillation terms, which may not be higher-order terms due to the low regularity of the numerical solution. Furthermore, the convergence of the algorithms is proved when bisection is used for the mesh refinements. Finally, numerical experiments are provided to verify the accuracy and efficiency of the ATG finite element algorithms compared to regular adaptive finite element algorithms and two-grid finite element algorithms [J. Xu, SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777]. [ABSTRACT FROM AUTHOR]

In this paper, we address the temporal energy growth associated with numerical approximations of the perfectly matched layer (PML) for Maxwell's equations in first order form. In the literature, several studies have shown that a numerical method which is stable in the absence of the PML can become unstable when the PML is introduced. We demonstrate in this paper that this instability can be directly related to numerical treatment of boundary conditions in the PML. First, at the continuous level, we establish the stability of the constant coefficient initial boundary value problem for the PML. To enable the construction of stable numerical boundary procedures, we derive energy estimates for the PML in the Laplace space. Second, we develop a high order accurate and stable numerical approximation for the PML using summation-by-parts finite difference operators to approximate spatial derivatives and weak enforcement of boundary conditions using penalties. By constructing analogous discrete energy estimates we show discrete stability and convergence of the numerical method. Numerical experiments verify the theoretical results. [ABSTRACT FROM AUTHOR]

Averaging techniques for finite element error control, occasionally called ZZ estimators for the gradient recovery, enjoy a high popularity in engineering because of their striking simplicity and universality: One does not even require a PDE to apply the nonexpensive postprocessing routines. Recently, averaging techniques have been mathematically proved to be reliable and efficient for various applications of the finite element method. This paper establishes a class of averaging error estimators for boundary integral methods. Symm's integral equation of the first kind with a nonlocal single-layer integral operator serves as a model equation studied both theoretically and numerically. We introduce four new error estimators which are proven to be reliable and efficient up to terms of higher order. The higher-order terms depend on the regularity of the exact solution. Several numerical experiments illustrate the theoretical results and show that the [normally unknown] error is sharply estimated by the proposed estimators, i.e., error and estimators almost coincide. [ABSTRACT FROM AUTHOR]