A multilevel homotopic adaptive finite element method is presented in this paper for convection dominated problems. By the homotopic method with respect to the diffusion parameter, the grids are iteratively adapted to better approximate the solution. Some new theoretic results and practical techniques for the grid adaptation are presented. Numerical experiments show that a standard finite eleme...

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 proposes a novel computational framework for improving realism and fidelity of finite element analysis simulations through experimental test data. The proposed scheme utilizes an artificial neural network to learn and compensate for the differences between a finite element analysis model simulation and corresponding experiment. The proposed computational methodology is poised to sign...

In the case that the domain has reentrant edges, the standard finite element method loses its global accuracy because of singularities on the boundary. To overcome this difficulty, FOSLL* is applied in this paper. FOSLL* is a methodology for solving PDEs using the dual operator. Here, a modified FOSLL* method is developed that employs a partially weighted functional and allows the use of a stan...

In this paper, we study adaptive finite element discretisation schemes for a class of parameter estimation problem. We propose to use adaptive multi-meshes in developing efficient algorithms for the estimation problem. We derive equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The err...

A new positive definite expanded mixed finite element method is proposed for parabolic partial integrodifferential equations. Compared to expanded mixed scheme, the new expanded mixed element system is symmetric positive definite and both the gradient equation and the flux equation are separated from its scalar unknown equation. The existence and uniqueness for semidiscrete scheme are proved an...

In this note, Finite Element Method is applied to solve the symmetric t-hyperbolic system with dissipative boundary condition and its stability is proved. In two-dimensional space, complex program is developed for the numerical solution of the mixed problem in simple connected region on the uniform grid. Delphi-7 is used for the code of the complex program. Numerical results are in line with th...

We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with ...

The Landau-Lifshitz-Gilbert Equation describes the dynamics of ferromagnetism, where strong nonlinearity and non-convexity are hard to tackle. Based on the work of S.Bartels and A.Prohl "Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation" ([4]), we present in this report a fully implicit finite element scheme with exchange and magnetostriction. We verify u...

A new adaptive finite element method for solving the Stokes equations is developed, which is shown to converge with the best possible rate. The method consists of 3 nested loops. The outermost loop consists of an adaptive finite element method for solving the pressure from the (elliptic) Schur complement system that arises by eliminating the velocity. Each of the arising finite element problems...