In this article, we discuss the numerical solution of the Stokes and Navier-Stokes equations completed by nonlinear slip boundary conditions of friction type in two and three dimensions. To solve the Stokes system, we first reduce the related variational inequality into a saddle point-point problem for a well chosen augmented Lagrangian. To solve this saddle point problem we suggest an alternat...

We first propose an adaptive immersed finite element method based on the a posteriori error estimate for solving elliptic equations with non-homogeneous boundary conditions in general Lipschitz domains. The underlying finite element mesh need not fit the boundary of the domain. Optimal a priori error estimate of the proposed immersed finite element method is proved. The immersed finite element ...

High-order accurate methods for convection-dominated problems have the potential to reduce the computational effort required for a given order of solution accuracy. The state of the art in this field is more advanced for Eulerian methods than for semi-Lagrangian (SLAG) methods. In this paper, we introduce a new SLAG method that is based on combining the modified method of characteristics with a...

In this dissertation a moving mesh method finite element method is used to approximate moving boundary solutions to the shallow water equations. An Arbitrary Lagrangian Eulerian method is applied to an existing finite element scheme. Some exact solutions to the shallow water equations in a parabolic basin are shown for comparison. An investigation is conducted into the accuracy of the method. I...

This paper introduces a non-oscillatory method, the finite element flux-corrected transport (FE-FCT) method for spallation studies. This method includes the implementation of a one-dimensional FCT algorithm into a total Lagrangian finite element method. Consequently, the FE-FCT method can efficiently eliminate fluctuations behind shock wave fronts without smearing them. In multidimensional simu...

In this article, we propose and analyze an effective Hessian recovery strategy for the Lagrangian finite element method of arbitrary order. We prove that the proposed Hessian recovery method preserves polynomials of degree k + 1 on general unstructured meshes and superconverges at a rate of O(hk) on mildly structured meshes. In addition, the method is proved to be ultraconvergent (two orders hi...

By minimizing the total potential energy function and deploying the virtual work principle, a higher-order stiffness matrix is achieved. This new tangent stiffness matrix is used to solve the frame with geometric nonlinear behavior. Since authors’ formulation takes into account the higher-order terms of the strain vector, the convergence speed of the solution process will increase. In fac...

We present the computation of lagrangian-based flow characterization measures for time-dependent, deformable-wall, finite-element blood flow simulations. Applicability of the algorithm is demonstrated in a fluid-structure interaction simulation of blood flow through a total cavopulmonary connection (Fontan procedure), and results are compared with a rigid-vessel simulation. Specifically, we rep...