In this paper, we propose a type of adaptive multigrid method for eigenvalue problem based on the multilevel correction method and adaptive multigrid method. Different from the standard adaptive finite element method applied to eigenvalue problem, with our method we only need to solve a linear boundary value problem on each adaptive space and then correct the approximate solution by solving a low dimensional eigenvalue problem. Further, the involved boundary value problems are solved by some adaptive multigrid iteration steps. The proposed adaptive algorithm can reach the same accuracy as the standard adaptive finite element method for eigenvalue problem but evidently reduces the computational work. In addition, the corresponding convergence and optimal complexity analysis are derived theoretically and numerically, respectively.

We consider the Cauchy problem for a second-order evolutionary equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the product of the operator $A$ by its conjugate operator $A^∗$. Time approximations are implemented so that the transition to a new level in time is associated with a separate solution of problems for operators $A$ and $A^∗$, not their products. The construction of unconditionally stable schemes is based on general results of the theory of stability (well-posedness) of operator-difference schemes in Hilbert spaces and is associated with the multiplicative perturbation of the problem operators, which lead to stable implicit schemes. As an example, the problem of the dynamics of a thin plate on an elastic foundation is considered.

A new primal-dual weak Galerkin (PDWG) finite element method is introduced and analyzed for the ill-posed elliptic Cauchy problems with ultra-low regularity assumptions on the exact solution. The Euler-Lagrange formulation resulting from the PDWG scheme yields a system of equations involving both the primal equation and the adjoint (dual) equation. The optimal order error estimate for the primal variable in a low regularity assumption is established. A series of numerical experiments are illustrated to validate effectiveness of the developed theory.

In the present paper, we propose and analyze a nonlinear fractional-order SEIR (susceptible-exposed-infected-recovered) epidemic model to transmit HIV. The fixed points of the model and their stability results are obtained. Using the fractional derivatives, we relied on the Caputo fractional derivative. Further, we employed the homotopy analysis method (HAM) to get an approximate solution of the dynamic fractional derivatives of the model. The purpose of using HAM as a solution technique is its reliability, easy to handle, that utilizes a simple process to adjust and control the convergence region of the obtained infinite series solution. It uses an auxiliary parameter and allows to obtain a one-parametric family of explicit series solutions. Firstly, several $h$-curves are plotted to demonstrate the regions of convergence, then the residual and square residual errors are obtained for different values of these regions. In the end, numerical solutions are presented for various iterations to show the accuracy of the HAM. Besides, the convergence theorem of HAM is also proved. The obtained results show the effectiveness and strength of the applied HAM on the proposed fractional-order SEIR model. Also, from the sensitivity analysis results, it is seen that the parameters $\mu$ and $\sigma$ are more sensitive than $\epsilon$ and $\rho$ in disease transmission.

Peritectic crystallization is a process in which the solid phase precipitated in the form of solid solution reacts with the liquid phase to form another solid phase. The process can be described by a phase field model where two continuous phase variables, $\phi$ and $\psi$, are introduced to distinguish the three different phases. We discretize the time variable with a scalar auxiliary variable (SAV) method that can ensure the unconditional energy stability. Moreover, the SAV method only requires solving a linear system at each time step and therefore reduces the computational complexity. The space variables in a two-dimensional region are discretized by an operator splitting method equipped with a high order compact finite difference formulation. This approach is effective and convenient since only a series of one-dimensional problems need to be solved at each step. We prove the unconditional energy stability theoretically and test the order of convergence and energy stability through numerical experiments. Simulations of peritectic solidification demonstrate the patterns formed during the process.

We consider a moving boundary problem with kinetic condition that describes the diffusion of solvent into rubber and study semi-discrete finite element approximations of the corresponding weak solutions. We report on both a priori and a posteriori error estimates for the mass concentration of the diffusants, and respectively, for the a priori unknown position of the moving boundary. Our working techniques include integral and energy-based estimates for a nonlinear parabolic problem posed in a transformed fixed domain combined with a suitable use of the interpolation-trace inequality to handle the interface terms. Numerical illustrations of our FEM approximations are within the experimental range and show good agreement with our theoretical investigation. This work is a preliminary investigation necessary before extending the current moving boundary modeling to account explicitly for the mechanics of hyperelastic rods to capture a directional swelling of the underlying elastomer.

This paper develops a recovery-based a posteriori error estimation for elliptic interface problems based on partially penalized immersed finite element (PPIFE) methods. Due to the low regularity of solution at the interface, standard gradient recovery methods cannot obtain superconvergent results. To overcome this drawback a new gradient recovery method is proposed that applies superconvergent cluster recovery (SCR) operator on each subdomain and weighted average (WA) operator at recovering points on the approximated interface. We prove that the recovered gradient superconverges to the exact gradient at the rate of $O(h^{1.5}).$ Consequently, the proposed method gives an asymptotically exact a posteriori error estimator for the PPIFE methods and the adaptive algorithm. Numerical examples show that the error estimator and the corresponding adaptive algorithm are both reliable and efficient.