In this paper, we propose a new stabilized linear finite element method for solving reaction-convection-diffusion equations with arbitrary magnitudes of reaction and diffusion. The key feature of the new method is that the test function in the stabilization term is taken in the adjoint-operator-like form $-\varepsilon \Delta v- ({\mbf a}\cdot\nabla v)/\gamma+\sigma v$, where the parameter $\gamma$ is appropriately designed to adjust the convection strength to achieve high accuracy and stability. We derive the stability estimates for the finite element solutions and establish the explicit dependence of $L^2$ and $H^1$ error bounds on the diffusivity, modulus of the convection field, reaction coefficient and the mesh size. The analysis shows that the proposed method is suitable for a wide range of mesh P\'{e}clet numbers and mesh Damk\"{o}hler numbers. More specifically, if the diffusivity $\varepsilon$ is sufficiently small with $\varepsilon < \Vert{\mbf a}\Vert h$ and the reaction coefficient $\sigma$ is large enough such that $\Vert{\mbf a}\Vert < \sigma h$, then the method exhibits optimal convergence rates in both $L^2$ and $H^1$ norms. However, for a small reaction coefficient satisfying $\Vert{\mbf a}\Vert \ge \sigma h$, the method behaves like the well-known streamline upwind/Petrov-Galerkin formulation of Brooks and Hughes. Several numerical examples exhibiting boundary or interior layers are given to demonstrate the high performance of the proposed method. Moreover, we apply the developed method to time-dependent reaction-convection-diffusion problems and simulation results show the efficiency of the approach.

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We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX − XD − AX + B = 0, with M ≡ [D,−C;−B,A] ∈ R(n1+n2)×(n1+n2) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A−1u, A−⊤u, D−1v, and D−⊤v computable in O(n) complexity (with n = max{n1, n2}), for some vectors u and v, and B, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu [Numer. Math., 103 (2006), pp. 392–412] is adapted, with the appropriate applications of the Sherman–Morrison– Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.

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In this paper, we develop an energy stable fully discrete discontinuous Galekin (DG) finite element method for the thin film epitaxy problem. Based on the method of lines, we construct and prove the energy stability of the spatial semi-discrete DG scheme firstly. To avoid the strict time step restriction of the explicit time integration method, the first order convex splitting method is used to get an unconditionally stable fully discrete DG method, which is a linearly implicit scheme for this nonlinear problem. The energy stability of the fully discrete convex splitting DG scheme is also proved. To improve the temporal accuracy, spectral deferred correction (SDC) method is adapted to achieve the high order accuracy in both time and space. Combining with the convex splitting method, the SDC method can be linearly solvable, high order accurate and stable in our numerical tests. These advantages ensure that the resulting fully discrete DG scheme is efficient to perform the long time simulation of the thin film epitaxy model. Numerical experiments of the accuracy and long time simulation show the capability and efficiency of the method.