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In this article, we consider the application of three popular domain decomposition methods to Lagrange-type nonconforming finite element discretizations of scalar, self-adjoint, second order elliptic equations. The additive Schwarz method of Dryja and Widlund, the vertex space method of Smith, and the balancing method of Mandel applied to nonconforming elements are shown to converge at a rate no worse than their applications to the standard conforming piecewise linear Galerkin discretization. Essentially, the theory for the nonconforming elements is inherited from the existing theory for the conforming elements with only modest modification by constructing an isomorphism between the nonconforming finite element space and a space of continuous piecewise linear functions.

The applicants have developed a finite element based approach for the solution of three-dimensional compressible flows. The procedure enables flow solutions to be obtained on tetrahedral discretizations of computational domains of complex form. A further development was the incorporation of a solution adaptive mesh strategy in which the adaptivity is achieved by complete remeshing of the solution domain. During the previous year, the applicants were working with the Advanced Aerodynamics Concepts Branch at NASA Ames Research Center with an implementation of the basic meshing and solution procedure. The objective of the work to be performed over this twelve month period was the transfer of the adaptive mesh technology and also the undertaking of basic research into alternative flow algorithms for the Euler equations on unstructured meshes.

A general numerical strategy for solving partial differential equations and other functional problems by cycling between coarser and finer levels of discretization is described. Optimal discretization schemes are provided together with very fast general solvers. It is described in terms of finite element discretizations of general nonlinear minimization problems. The basic processes (relaxation sweeps, fine-grid-to-coarse-grid transfers of residuals, coarse-to-fine interpolations of corrections) are directly and naturally determined by the objective functional and the sequence of approximation spaces. The natural processes, however, are not always optimal. Concrete examples are given and some new techniques are reviewed. Including the local truncation extrapolation and a multilevel procedure for inexpensively solving chains of many boundary value problems, such as those arising in the solution of time-dependent problems.

The goal of this work is to develop robust schemes for the simulation of crack propagation using adaptive finite element analysis. Here we address the main problem of estimation of the error in the engineering quantity of interest (e.g., stress intensity factors for the crack, the stresses in the hot spots). We showed that the error in the quantity of interest can be estimated by employing the standard error estimators, for the energy norm of the error, which are existing in several commercial finite element programs. We have developed a computer based approach for checking the quality of the error estimators used in practice and we have also developed error estimators which are robust for meshes used in engineering practice. This work has led to an innovative approach for adaptive mesh refinement to obtain the quantity of interest with prescribed tolerance.

A second order finite difference method has been developed for elliptic interface problems that involve discontinuous coefficients, singular source terms, non-smooth or even discontinuous coefficients across an arbitrary interface. The method is based on Cartesian grids and coupled with a multigrid solver. The new method will preserve the discrete maximum principle. The convergence of the new method has been proved. Two different finite element spaces and corresponding Galerkin methods for elliptic interface problems using Cartesian grids have been developed. Some theoretical results about these finite element methods are also obtained. We believe that we are the first to develop these new finite element spaces over Cartesian grids. Some public subroutines of fast solvers for Helmholtz and Poisson equations on irregular domains either exterior or interior with various boundary conditions have been developed and made public In collaboration with J. K. Hunter of UC Davis and H. K. Zhao of UC Irvine, we formulate a model for the spreading on a surface of a drop that deposits an autophobic monolayer of surfactant We present numerical solutions of the model equations using an immersed interface method and a level set method.

A second order finite difference method has been developed for elliptic interface problems that involve discontinuous coefficients, singular source terms, non-smooth or even discontinuous coefficients across an arbitrary interface. The method is based on Cartesian grids and coupled with a multigrid solver. The new method will preserve the discrete maximum principle. The convergence of the new method has been proved. Two different finite element spaces and corresponding Galerkin methods for elliptic interface problems using Cartesian grids have been developed. Some theoretical results about these finite element methods are also obtained. We believe that we are the first to develop these new finite element spaces over Cartesian grids. Some public subroutines of fast solvers for Helmholtz and Poisson equations on irregular domains either exterior or interior with various boundary conditions have been developed and made public In collaboration with J. K. Hunter of UC Davis and H. K. Zhao of UC Irvine, we formulate a model for the spreading on a surface of a drop that deposits an autophobic monolayer of surfactant We present numerical solutions of the model equations using an immersed interface method and a level set method.

A second order finite difference method has been developed for elliptic interface problems that involve discontinuous coefficients, singular source terms, non-smooth or even discontinuous coefficients across an arbitrary interface. The method is based on Cartesian grids and coupled with a multigrid solver. The new method will preserve the discrete maximum principle. The convergence of the new method has been proved. Two different finite element spaces and corresponding Galerkin methods for elliptic interface problems using Cartesian grids have been developed. Some theoretical results about these finite element methods are also obtained. We believe that we are the first to develop these new finite element spaces over Cartesian grids. Some public subroutines of fast solvers for Helmholtz and Poisson equations on irregular domains either exterior or interior with various boundary conditions have been developed and made public In collaboration with J. K. Hunter of UC Davis and H. K. Zhao of UC Irvine, we formulate a model for the spreading on a surface of a drop that deposits an autophobic monolayer of surfactant We present numerical solutions of the model equations using an immersed interface method and a level set method.

A second order finite difference method has been developed for elliptic interface problems that involve discontinuous coefficients, singular source terms, non-smooth or even discontinuous coefficients across an arbitrary interface. The method is based on Cartesian grids and coupled with a multigrid solver. The new method will preserve the discrete maximum principle. The convergence of the new method has been proved. Two different finite element spaces and corresponding Galerkin methods for elliptic interface problems using Cartesian grids have been developed. Some theoretical results about these finite element methods are also obtained. We believe that we are the first to develop these new finite element spaces over Cartesian grids. Some public subroutines of fast solvers for Helmholtz and Poisson equations on irregular domains either exterior or interior with various boundary conditions have been developed and made public In collaboration with J. K. Hunter of UC Davis and H. K. Zhao of UC Irvine, we formulate a model for the spreading on a surface of a drop that deposits an autophobic monolayer of surfactant We present numerical solutions of the model equations using an immersed interface method and a level set method.

In the first part of this paper, we establish a conditional optimality result for an adaptive mixed finite element method for the stationary Stokes problem discretized by the standard Taylor-Hood elements, under the assumption of the so-called general quasi-orthogonality. Optimality is measured in terms of a modified approximation class defined through the total error, as is customary since the seminal work of Cascon, Kreuzer, Nochetto and Siebert. The second part of the paper is independent of optimality results, and concerns interrelations between the modified approximation classes and the standard approximation classes (the latter defined through the energy error). Building on the tools developed in the papers of Binev, Dahmen, DeVore, and Petrushev, and of Gaspoz and Morin, we prove that the modified approximation class coincides with the standard approximation class, modulo the assumption that the data is regular enough in an appropriate scale of Besov spaces.

A finite element method, developed for static analysis of composite plates, was enhanced to be used with sandwich plates. The theory considers geometric nonlinearity and transverse shear. Furthermore, a new postprocessor was written to check for initial failure using the maximum stress criteria. It also includes a procedure for evaluating transverse normal stresses by enforcing equilibrium. The code modifications for sandwiches were verified by by comparing finite element solutions to closed-form sandwich theories. Both methods showed good correlation. In addition, comparisons between one type of composite plate and a similar sandwich plate found that the sandwich had better specific stiffness for various geometries. Finally, quasi-static models of low-velocity impacts on sandwich plates succeeded in predicting incipient damage due to core crushing in the impact region, but they were inconsistent in estimating the actual load levels for initial failure, based on experiments from other research.

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger--Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

We consider a two-stage numerical procedure for imaging of objects buried in dry sand using time-dependent backscattering experimental radar measurements. These measurements are generated by a single point source of electric pulses and are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. Our imaging problem is formulated as the inverse problem of the reconstruction of the spatially distributed dielectric permittivity $\varepsilon_\mathrm{r}\left(\mathbf{x}\right), \ \mathbf{x}\in \mathbb{R}^{3}$, which is an unknown coefficient in Maxwell's equations. On the first stage an approximately globally convergent method is applied to get a good first approximation for the exact solution. On the second stage a local adaptive finite element method is applied to refine the solution obtained on the first stage. The two-stage numerical procedure results in accurate imaging of all three components of interest of targets: shapes, locations and refractive indices. In this paper we briefly describe methods and present new reconstruction results for both stages.

For a family of stabilized mixed finite element methods for the Stokes equations a complete a priori and a posteriori error analysis is given.

In this paper, we derive a priori error estimates for a class of interior penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE) functions for a classic second-order elliptic interface problem. The error estimation shows that these methods can converge optimally in a mesh-dependent energy norm. The combination of IFEs and DG formulation in these methods allows local mesh refinement in the Cartesian mesh structure for interface problems. Numerical results are provided to demonstrate the convergence and local mesh refinement features of these DG-IFE methods.

Fluid flow applications can involve a number of coupled problems. One is the simulation of free-surface flows, which require the solution of a free-boundary problem. Within this problem, the governing equations of fluid flow are coupled with a domain deformation approach. This work reviews five of those approaches: interface tracking using a boundary-conforming mesh and, in the interface capturing context, the level-set method, the volume-of-fluid method, particle methods, as well as the phase-field method. The history of each method is presented in combination with the most recent developments in the field. Particularly, the topics of extended finite elements (XFEM) and NURBS-based methods, such as Isogeometric Analysis (IGA), are addressed. For illustration purposes, two applications have been chosen: two-phase flow involving drops or bubbles and sloshing tanks. The challenges of these applications, such as the geometrically correct representation of the free surface or the incorporation of surface tension forces, are discussed.

The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasi-orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.

We consider a stabilized nonconforming finite element method for data assimilation in incompressible flow subject to the Stokes' equations. The method uses a primal dual structure that allows for the inclusion of nonstandard data. Error estimates are obtained that are optimal compared to the conditional stability of the ill-posed data assimilation problem.

This is the second paper on the study of gradient recovery for elliptic interface problem. In our previous work [H. Guo and X. Yang, 2016, arXiv:1607.05898], we developed gradient recovery finite element method based on body-fitted mesh. In this paper, we propose new gradient recovery methods based on two immersed interface finite element methods: symmetric and consistent immersed finite method [H. Ji, J. Chen and Z. Li, J. Sci. Comput., 61 (2014), 533--557] and Petrov-Galerkin immersed finite element method [T.Y. Hou, X.-H. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185--205, and S. Hou and X.-D. Liu, J. Comput. Phys., 202 (2005), 411--445]. Compared to body-fitted mesh based gradient recover methods, immersed finite element methods provide a uniform way of recovering gradient on regular meshes. Numerical examples are presented to confirm the superconvergence of both gradient recovery methods. Moreover, they provide asymptotically exact a posteriori error estimators for both immersed finite element methods.

This paper introduces new discretization schemes for time-harmonic Maxwell equations in a connected domain by using the weak Galerkin (WG) finite element method. The corresponding WG algorithms are analyzed for their stability and convergence. Error estimates of optimal order in various discrete Sobolev norms are established for the resulting finite element approximations.

We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.

In this paper, we develop two classes of mixed finite element schemes for stationary magnetohydrodynamics (MHD) models, one using the magnetic field $B$ and the electric field $ E$ as the discretization variables while the other using $ B$ and the current density $j$ as the discretization variables. We show that the Gauss's law for magnetic field, namely $\nabla\cdot {B}=0$, and the energy law for the entire system are exactly preserved in both of these finite element schemes. Based on some new basic estimates for $H^{h}(\mathrm{div})$, we show that both of these new finite element schemes are well-posed respectively under some conditions for the $ B$-$ E$ formulation but no conditions for the $ B$-$ j$ formulation. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of Picard iterations under some conditions.

Book Source: Digital Library of India Item 2015.193316 dc.contributor.author: Amiya Kumar Pani dc.date.accessioned: 2015-07-08T02:30:55Z dc.date.available: 2015-07-08T02:30:55Z dc.date.digitalpublicationdate: 2005-08-27 dc.identifier.barcode: 1990010086728 dc.identifier.origpath: /rawdataupload/upload/0086/728 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/193316 dc.description.scannerno: 14 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 180 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Mechanical Engineering In General. Nuclear Technology. Electrical Engineering. Machinery dc.title: Finite Element Methods For Nonlinear Free Boundary Problems

At the start, three and a half years ago, the finite fluid element method (Section 1) was the only one under consideration. Things went badly with that method, and progress with its implementation went far more slowly than we ever anticipated. As a result, by the end of the first year, a second method, which originally was developed for a check on results of the first, had become by far the more promising of the two. Most of this report (Sections 3-6) describes progress we have made with the application of biased differences (Section 2) to a variety of fairly difficult problems of numerical fluid mechanics. Finally, in the last six months of the period covered by this report, the major difficulties with finite fluid elements were overcome, so it became possible to begin a comparison of the two methods (Section 6). Preliminary indications are that both methods are reliable, and both are considerably more efficient than a third method with which they have been compared. Section 7 is a report of progress with a boundary integral method that is not closely related to the others except by being numerical, and Section 8 is a description of the kinds of graphical software we had to develop for interpretation of our numerical computations.

The primary purpose of this paper is to compare the accuracy and performance of two numerical approaches to solving systems of partial differential equations. These equations are posed on adjoining domains sharing boundary conditions on a common boundary interface in the important case when the meshes used on the two domains are non-matching across the interface. The first widely used approach is based on a finite volume method employing ad hoc projections to relate approximations on the two domains across the interface. The second approach uses the mathematically-founded mortar mixed finite element method. To quantify the performance, we use a goal-oriented a-posteriori error estimate that quantifies various aspects of discretization error to the overall error. While the performance difference may be not a surprise in some cases, we believe that there is a perception in part of the scientific community concerned with multiphysics systems that if the solution is smooth near the interface, then there is little effect from varying the coupling technique. We find that, on the contrary, the error associated with ad hoc coupling approaches may be large in practical situations. Moreover, we also show that mortar methods can be used with black box component solves, thus permitting an efficient and practical implementation also within legacy codes.

This report presents in detail the application of an energy method to describe a class of delamination growth problems found in polymer-based composite laminates. The energy method is derived from the general concept of the classification fracture mechanics. When used judiciously and with proper interpretation of the physical factors pertinent to the crack growth process at the lamina(ply) level, the method provides an analytical simulation for the delamination process in laminates subjected to monotonic loading and/or cyclic loading, both in tension and in compression. Case studies involving one-dimensional growth (e.g. uniform free edge delamination) and two-dimensional delamination growth (contour delamination) have been performed along with laboratory tests using laminates made of AS-3501-06 graphite-epoxy system. Excellent correlation between experimental and predicted results has been achieved. (Author)

In this paper we consider the approximate solution of a class of second order elliptic equations with rough coefficients. Problem of the type considered arise in the analysis of unidirectional composites, where the coefficients represent the properties of the material. We present several methods for this class of problem, and show that they have the same accuracy as usual methods have for problem with smooth coefficients. We refer to the methods as special finite elements methods because they are of finite element type but employ special shape functions, chosen to effectively model the unknown solution.

The objective of this Workshop was to consider the current position of applying the finite element method to applications involving composite materials. It was expected that issues needing attention would be identified. Some technical conclusions were as follows: (1) The actual material must be modelled; (2) Properties must be appropriate to the scale of the analysis; (3) The system must allow for correct orientation of the material; (4) It should be possible to represent (ply) discontinuities; (5) It should be possible to incorporate through-thickness modelling; (6) Specific formulation is needed to allow for iteration; (7) It should be possible to include material and geometric non-linearity; (8) Include relevant critical mechanical properties; (9) System must be able to identify failure mode(s); and (10) Consider using spreadsheets to interface input data with FE system users need more, and better, communication with software vendors establish a system (benchmarks) for verifying results.

A summary of the 2nd Japan-US Symposium on Finite Element Methods in Large Scale Computational Fluid Dynamics, conducted Mar. 14-16, 1994 at Chuo University in Tokyo, Japan is presented. Abstracts of all symposium presentations are included. This report is based upon information collected via symposium attendance, review of the symposium proceedings, and a post symposium conversation with Professor Mutsuto Kawahara, Department of Civil Engineering, Chuo University, who served as the Japanese co-chair of the symposium organizing committee. (AN)

A summary of the 2nd Japan-US Symposium on Finite Element Methods in Large Scale Computational Fluid Dynamics, conducted Mar. 14-16, 1994 at Chuo University in Tokyo, Japan is presented. Abstracts of all symposium presentations are included. This report is based upon information collected via symposium attendance, review of the symposium proceedings, and a post symposium conversation with Professor Mutsuto Kawahara, Department of Civil Engineering, Chuo University, who served as the Japanese co-chair of the symposium organizing committee. (AN)

We construct and analyze a preconditioner of the linear elastiity system discretized by conforming linear finite elements in the framework of the auxiliary space method. The auxiliary space preconditioner is based on discretization of a scalar elliptic equation with Generalized Finite Element Method.

We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique. Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA.

This report discusses the reliability of finite element analysis techniques in place of structural test in the context of airworthiness assessment. Using an example as illustration, the paper shows that in order to obtain the meaningful results, the analysis must be performed with extreme care by an experienced analyst. The reliance on the finite element analysis alone may result in serious misdiagnosis. The verification and the certification of programs and the certification of users are discussed. Some of the most important tests for validation of finite element are presented. Keywords: NATO furnished; Aviation safety; Safety engineering; Reliability; Computerized simulation.

This report discusses the reliability of finite element analysis techniques in place of structural test in the context of airworthiness assessment. Using an example as illustration, the paper shows that in order to obtain the meaningful results, the analysis must be performed with extreme care by an experienced analyst. The reliance on the finite element analysis alone may result in serious misdiagnosis. The verification and the certification of programs and the certification of users are discussed. Some of the most important tests for validation of finite element are presented. Keywords: NATO furnished; Aviation safety; Safety engineering; Reliability; Computerized simulation.

This report discusses the reliability of finite element analysis techniques in place of structural test in the context of airworthiness assessment. Using an example as illustration, the paper shows that in order to obtain the meaningful results, the analysis must be performed with extreme care by an experienced analyst. The reliance on the finite element analysis alone may result in serious misdiagnosis. The verification and the certification of programs and the certification of users are discussed. Some of the most important tests for validation of finite element are presented. Keywords: NATO furnished; Aviation safety; Safety engineering; Reliability; Computerized simulation.

Unfitted finite element methods have a great potential for large scale simulations, since avoid the generation of body-fitted meshes and the use of graph partitioning techniques, two main bottlenecks for problems with non-trivial geometries. However, the linear systems that arise from these discretizations can be much more ill-conditioned, due to the so-called small cut cell problem. The state-of-the-art approach is to rely on sparse direct methods, which have quadratic complexity and thus, are not well-suited for large scale simulations. In order to solve this situation, in this work we investigate the use of domain decomposition preconditioners (balancing domain decomposition by constraints) for unfitted methods. We observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior. As a result, we propose an enhancement of the classical BDDC methods based on 1) a modified (stiffness) weighting operator and 2) an improved definition of the coarse degrees of freedom in the definition of the preconditioner . These changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids. A complete set of complex 3D numerical experiments show the good performance of the proposed preconditioners.

This thesis is concerned with the use of multi-level methods to solve the linear systems arising from finite element discretizations of elliptic equations. In all, three multi-level methods are considered. The first of these is applicable only to quasi-uniform grids, but is simpler than other algorithms considered in previous theoretical work. The other two algorithms are applicable to both quasi-uniform grids, and locally refined grids, those grids on which the size of the largest and smallest elements many differ by an arbitrarily large factor. All three algorithms are asymptotically optimal, producing good solutions in O(N) operations on a finite element grid with N elements. These asymptotically optimal complexity bounds for the last two algorithms are the first such bounds for multi-level methods on locally refined grids. One of these algorithms achieves this O(N) complexity bound under weaker than expected conditions on the dimensions of the finite element spaces used by the algorithm.

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

The transient heat conduction equation with a nonlinear conductivity is solved using three finite difference schemes and two finite element schemes. The schemes are compared for speed and accuracy against an analytic solution. The finite element methods are shown to be the most accurate and one, a lumped mass method, is comparable in speed to the finite difference methods. The lumped mass finite element method is shown to be the clear choice from among the methods tested in solving the test problem. (Author)

This annual status report presents the results of work performed during the fourth year of the 3-D Inelastic Analysis Methods for Hot Section Components program (NASA Contract NAS3-23697). The objective of the program is to produce a series of new computer codes permitting more accurate and efficient 3-D analysis of selected hot section components, i.e., combustor liners, turbine blades and turbine vanes. The computer codes embody a progression of math models and are streamlined to take advantage of geometrical features, loading conditions, and forms of material response that distinguish each group of selected components. Volume 1 of this report discusses the special finite element models developed during the fourth year of the contract.

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

The Adaptive Hydraulics (ADH) model is an adaptive finite element method to simulate three-dimensional Navier-Stokes flow, unsaturated and saturated groundwater flow, overland flow, and two- or three-dimensional shallow-water flow and transport. In the shallow-water flow and transport, especially involving multispecies transport, the water depth (h), the product of water depth and velocities (uh and vh), as well as water depth and chemical concentration (hc) are dependent variables of fluid-motion simulations and are often solved at various times. It is important for the numerical model to predict accurate water depth, velocity fields, and chemical distribution, as well as conserve mass, especially for water quality applications. Solution accuracy depends highly on mesh resolution. Adaptive mesh refinement (AMR), particularly the h-refinement, is often used to add new nodes in the region where they are needed and to remove others where they are no longer required during the simulation. The AMR is proven to optimize the performance of a computed solution. However, mass with gain or loss can occur when elements are merged due to removing a node at mesh coarsening. Therefore, we develop and implement the mass-conservative unrefinement algorithm to ensure the mass conserved in a merged element in which a node has been removed. This study describes the use of the Galerkin finite element method to redistribute mass to nodes comprising a merged element. The algorithm was incorporated into the ADH code. This algorithm minimizes mass error during the unrefinement process to conserve mass during the simulation for two-dimensional shallow-water flow and transport. The implementation neither significantly increases the computational time nor memory usage. The simulation was run with various numbers of processors. The results showed good scaling of solution time as the number of processors increases.

Elementary descriptions of finite element and finite difference methods are given while the finite volume method is briefly overviewed. Examples illustrating finite element and finite difference methods are worked out. Finally, comparisons of these methods between themselves and with some examples from literature are given. (AN)

We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in $\mathbb{R}^3$. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements with one order higher polynomial degree for the mapping. The tangent condition is weakly enforced using a penalization term. We derive error estimates that takes the approximation of both the geometry of the surface and the solution to the partial differential equation into account. We also present numerical results that verify our theoretical findings.

Our goal is to present an elementary approach to the analysis and programming of sparse grid finite element methods. This family of schemes can compute accurate solutions to partial differential equations, but using far fewer degrees of freedom than their classical counterparts. After a brief discussion of the classical Galerkin finite element method with bilinear elements, we give a short analysis of what is probably the simplest sparse grid method: the two-scale technique of Lin et al. (2001). We then demonstrate how to extend this to a multiscale sparse grid method which, up to choice of basis, is equivalent to the hierarchical approach, as described by, e.g., Bungartz and Griebel (2004). However, by presenting it as an extension of the two-scale method, we can give an elementary treatment of its analysis and implementation. For each method considered, we provide MATLAB code, and a comparison of accuracy and computational costs.

The comparative computational efficiencies of the finite element (FEM), boundary element (BEM), and hybrid boundary element-finite element (HBFEM) analysis techniques are evaluated for representative bounded domain interior and unbounded domain exterior problems in elastostatics. Computational efficiency is carefully defined in this study as the computer time required to attain a specified level of solution accuracy. The study found the FEM superior to the BEM for the interior problem, while the reverse was true for the exterior problem. The hybrid analysis technique was found to be comparable or superior to both the FEM and BEM for both the interior and exterior problems.

In this article, we propose and analyze an effective Hessian recovery strategy for the Lagrangian finite element of arbitrary order $k$. We prove that the proposed Hessian recovery preserves polynomials of degree $k+1$ on general unstructured meshes and superconverges at rate $O(h^k)$ on mildly structured meshes. In addition, the method preserves polynomials of degree $k+2$ on translation invariant meshes and produces a symmetric Hessian matrix when the sampling points for recovery are selected with symmetry. Numerical examples are presented to support our theoretical results.

In this paper, the authors devise a new discretization scheme for div-curl systems defined in connected domains with heterogeneous media by using the weak Galerkin finite element method. Two types of boundary value problems are considered in the algorithm development: (1) normal boundary condition, and (2) tangential boundary condition. A new variational formulation is developed for the normal boundary value problem by using the Helmholtz decomposition which avoids the computation of functions in the harmonic fields. Both boundary value problems are reduced to a general saddle-point problem involving the curl and divergence operators, for which the weak Galerkin finite element method is devised and analyzed. The novelty of the technique lies in the discretization of the divergence operator applied to vector fields with heterogeneous media. Error estimates of optimal order are established for the corresponding finite element approximations in various discrete Sobolev norms.

This article takes the form of a tutorial on the use of a particular class of mixed finite element methods, which can be thought of as the finite element extension of the C-grid staggered finite difference method. The class is often referred to as compatible finite elements, mimetic finite elements, discrete differential forms or finite element exterior calculus. We provide an elementary introduction in the case of the one-dimensional wave equation, before summarising recent results in applications to the rotating shallow water equations on the sphere, before taking an outlook towards applications in three-dimensional compressible dynamical cores.

We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of convergence for spatially semidiscrete approximations as well as a spatio-temporal discretization which is based on a linear implicit Euler-Maruyama method. In both cases we obtain optimal error estimates. The proofs are based on sharp integral versions of well-known error estimates for the corresponding deterministic linear homogeneous equation together with optimal regularity results for the mild solution of the SPDE. The results hold for different Galerkin methods such as the standard finite element method or spectral Galerkin approximations.