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We propose in this paper a new minimization algorithm based on a slightly modified version of the scalar auxialiary variable (SAV) approach coupled with a relaxation step and an adaptive strategy. It enjoys several distinct advantages over popular gradient based methods: (i) it is unconditionally energy diminishing with a modified energy which is intrinsically related to the original energy, and thus no parameter tuning is needed for stability; (ii) it allows the use of large step-sizes which can effectively accelerate the convergence rate. We also present a convergence analysis for some SAV based algorithms, which include our new algorithm without the relaxation step as a special case. We apply our new algorithm to several illustrative and benchmark problems and compare its performance with several popular gradient based methods. The numerical results indicate that the new algorithm is very robust, and its adaptive version usually converges significantly faster than those popular gradient decent based methods. [ABSTRACT FROM AUTHOR]

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]

This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem. Moreover, we prove the convergence of discretizations in two-dimensional situations. A numerical algorithm is devised that iteratively solves the discrete formulation. Numerical experiments show that optimal convex shapes are generally nonsmooth and that three-dimensional problems require an appropriate relaxation of the convexity condition. [ABSTRACT FROM AUTHOR]