Highly accurate finite element method for one-dimensional elliptic interface problems

In this study, we developed a neutron diffusion equation solver based on the finite element method for CAPP code. Three types of triangular finite elements and five types of rectangular depending on the order of the shape functions were implemented for 2-D application. Ten types of triangular prismatic finite elements and seventeen types of rectangular prismatic finite elements were also implemented for 3-D application. Two types of polynomial mapping from the master finite element to a real finite element were adopted for flexibility in dealing with complex geometry. They are linear mapping and iso-parametric mapping. In linear mapping, only the vertex nodes are used as the mapping points. In iso-parametric mapping, all the nodal points in the finite element are used as the mapping points, which enables the real finite elements to have curved surfaces. For the treatment of spatial dependency of cross-sections in the finite elements, three types of polynomial expansion of the cross-sections in the finite elements were implemented. They are constant, linear, and iso-parametric cross-section expansions. The power method with the Wielandt acceleration technique was adopted as the outer iteration algorithm. The BiCGSTAB algorithm with the ILU (Incomplete LU) decomposition preconditioner was used as the linear equation solver in the inner iteration. The neutron diffusion equation solver developed in this study was verified against two well known benchmark problems, IAEA PWR benchmark problem and OECD/NEA PBMR400 benchmark problem. Results of numerical tests showed that the solution converged to the reference solution as the finite elements are refined and as the order of the finite elements increases. Numerical tests also showed that the higher order finite element method is much efficient than lower order finite element method or finite difference method.

A high-order source removal finite element method for a class of elliptic interface problems

This document presents a new electromagnetic in-house parallel EM simulator named HOFEM (Higher Order Finite Element Method). The simulator makes use of some of the research developments on the area made within the Group of Radiofrequency, Electromagnetism, Microwaves and Antennas (GREMA) of the Universidad Carlos III de Madrid to which the authors belong to. HOFEM makes use of a weak formulation based on double curl vector wave equation discretized with the higher-order isoparametric curl-conforming tetrahedral (and prisms) finite element rigorous implementations of Nédélec's first family of elements. A non-standard mesh truncation technique, FE-IIEE (Finite Element - Iterative Integral Equation Evaluation) for open region problems provides an arbitrarily exact absorbing boundary condition while retaining the original sparse structure of the FEM matrices. The simulator provides a multi-platform (Linux & Windows) user friendly graphical user interface. Remote job submission to HPC clusters is supported by integrating an in-house software tool developed for this purpose.

Iterative solution of Helmholtz problem with high-order isogeometric analysis and finite element method at mid-range frequencies

High-order accurate finite element methods provide unique benefits for problems that have strong anisotropies and complicated geometries and for stiff equation systems that are coupled through large source terms, e.g. Lorentz force, collisions, or atomic reactions. Magnetized plasma simulations of realistic devices using the kinetic or the multi-fluid plasma models are examples that benefit from high-order accuracy. The multi-fluid plasma model only assumes local thermodynamic equilibrium within each fluid, e.g. ion and electron fluids for the two-fluid plasma model. The algorithm implements a discontinuous Galerkin method with an approximate Riemann solver to compute the fluxes of the fluids and electromagnetic fields at the computational cell interfaces. The multi-fluid plasma model has time scales on the order of the electron and ion cyclotron frequencies, the electron and ion plasma frequencies, the electron and ion sound speeds, and the speed of light. A general model for atomic reactions has been developed and is incorporated in the multi-fluid plasma model. The multi-fluid plasma algorithm is implemented in a flexible code framework (WARPX) that allows easy extension of the physical model to include multiple fluids and additional physics. The code runs on multi-processor machines and is being adapted with OpenCL to many-core systems, characteristic of the next generation of high performance computers. The algorithm is applicable to study advanced physics calculations of plasma dynamics including magnetic plasma confinement and astrophysical plasmas. The discontinuous Galerkin method has also been applied to solve the Vlasov-Poisson kinetic model. Recently, a blended finite element algorithm has been developed and implemented which exploits the expected physical behavior to apply either a discontinuous or continuous finite element representation, which improves computational efficiency without sacrificing accuracy.

Fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for a high-order finite element method (FEM) are proposed based on the fast discrete Fourier transform. They generalize logarithmically optimal Fourier algorithms for solving boundary value problems for Poisson-type equations on rectangular meshes to high-order FEM. The algorithms can be extended to the multidimensional case and can be applied to nonstationary problems.

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. We consider a new unfitted finite element method which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. The method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping are easy to implement. In this paper we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high order discretization. The theoretical results are confirmed in numerical experiments.

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new $Log$ marking strategy is proposed. The $Log$ marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes.

In this paper, a new high-order multiscale finite element method (MsFEM) is developed for elliptic problems with highly oscillating coefficients. The method is inspired by the MsFEM developed in [G. Allaire and R. Brizzi, Multiscale Model. Simul., 4 (2005), pp. 790--812], but a more explicit multiscale finite element space is constructed. The approximation space is nonconforming when an oversampling technique is used. We use a Petrov--Galerkin formulation suggested in [T. Y. Hou, X.-H. Wu, and Y. Zhang, Commun. Math. Sci., 2 (2004), pp. 185--205] to simplify the implementation and to improve the accuracy. The method is natural for high-order finite element methods used with the advantage of solving the coarse grained problem. We prove optimal error estimates in the case of periodically oscillating coefficients and support the findings by various numerical experiments.

Membrane forces acting on thin-walled box girders considering shear lag effect

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation

Effective condition number and superconvergence of the Trefftz method coupled with high order FEM for singularity problems

Multi-scale/level computer models are efficient for simulating solutions of partial differential equations on nontrivial geometries using a large number of modes. This is because fine-scale structures in the quantities of interest cannot, in practice, be resolved using a single-level finite element method (FEM) applied directly to the full model, or standard spectral basis functions are not appropriate for the nontrivial geometry and related constraints. The multi-scale/level FEM (MFEM) is based on first applying a FEM for a reduced model problem, obtained by restriction to either (i) several local subdomains or (ii) a dominant linear differential operator in the model. The resulting structure-aware precomputed solutions are then used as basis functions to simulate the full model in the MFEM framework. If these are eigenfunctions of a reduced model eigenvalue problem, such basis functions provide spectrally accurate approximations to the solution of the full model and the resulting method is called the spectral MFEM (SMFEM). In this article we develop, analyze, and implement time-space fully discrete implicit SMFEM (ISMFEM) algorithms for efficiently simulating a Ginzburg--Landau (GL) system modeling superconductivity on a class of superconducting surfaces $S$. The spatial linear differential operator in the time-dependent nonlinear GL model is the Schrödinger operator $( i \nabla + {A}_0)^2$ on $S$, with magnetic vector potential ${A}_0$. For our ISMFEM, the spectrally accurate basis functions are eigenfunctions of a reduced stationary linear model governed by $(i \nabla + {A}_0)^2$. In a recent work we developed, analyzed, and implemented an algorithm to compute these functions using an analytic-numeric hybrid method, in conjunction with a high-order FEM. Using these precomputed solutions as basis functions, the spectrally accurate algorithms developed and analyzed for our new GL computer models on nontrivial geometries are validated using both nonlinear and linearized implicit time discretizations.

Implementing and using high-order finite element methods