A locally divergence-free interior penalty method for two-dimensional curl-curl problems

An interior penalty method for certain two-dimensional curl-curl problems is investigated in this paper. This method computes the divergence-free part of the solution using locally divergence-free discontinuous P vector fields on graded meshes. It has optimal order convergence (up to an arbitrarily small e) for the source problem and the eigenproblem. Results of numerical experiments that corroborate the theoretical results are also presented. © 2008 Society for Industrial and Applied Mathematics.

Scaling-robust built-in a posteriori error estimation for discontinuous least-squares finite element methods

A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions for the solution of the Poisson model problem in 2D with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables a natural weighting of the jump in the least-squares functional in terms of the mesh size. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees

Comparison results for the Stokes equations

This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main comparison result is that the error of the P2-P0-FEM is a lower bound to the error of the Bernardi-Raugel (or reduced P2-P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs. Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition

Analysis and finite element approximations of stochastic optimal control problems constrained by stochastic elliptic partial differential equations

In this thesis we study mathematically and computationally optimal control problems for stochastic elliptic partial differential equations. The control objective is to minimize the expectation of a tracking cost functional, and the control is of the deterministic, distributed type. The main analytical tool is the Wiener-Ito chaos or the Karhunen-Loeve expansion. Mathematically, we prove the existence of an optimal solution; we establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations; we represent the stochastic functions in their Wiener-Ito chaos expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the optimality system through the discretizations of the probability space and the spatial space by the finite element method; we also derive error estimates in terms of both types of discretizations. Finally, we present some results of numerical experiments

Multigrid methods for Maxwell\u27s equations

In this work we study finite element methods for two-dimensional Maxwell\u27s equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell\u27s equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments

High-order discontinuous Galerkin methods for incompressible flows

Aquesta tesi doctoral proposa formulacions de Galerkin discontinu (DG) d'alt ordre per fluxos viscosos incompressibles. Es desenvolupa un nou mètode de DG amb penalti interior (IPM-DG), que condueix a una forma feble simètrica i coerciva pel terme de difusió, i que permet assolir una aproximació espacial d'alt ordre. Aquest mètode s'aplica per resoldre les equacions de Stokes i Navier-Stokes. L'espai d'aproximació de la velocitat es descompon dins de cada element en una part solenoidal i una altra irrotacional, de manera que es pot dividir la forma dèbil IPM-DG en dos problemes desacoblats. El primer permet el càlcul de les velocitats i de les pressions híbrides, mentre que el segon calcula les pressions en l'interior dels elements. Aquest desacoblament permet una reducció important del número de graus de llibertat tant per velocitat com per pressió. S'introdueix també un paràmetre extra de penalti resultant en una formulació DG alternativa per calcular les velocitats solenoidales, on les pressions no apareixen. Les pressions es poden calcular com un post-procés de la solució de les velocitats. Es contemplen altres formulacions DG, com per exemple el mètode Compact Discontinuous Galerkin, i es comparen al mètode IPM-DG. Es proposen mètodes implícits de Runge-Kutta d'alt ordre per problemes transitoris incompressibles, permetent obtenir esquemes incondicionalment estables i amb alt ordre de precisió temporal. Les equacions de Navier-Stokes incompressibles transitòries s'interpreten com un sistema de Equacions Algebraiques Diferencials, és a dir, un sistema d'equacions diferencials ordinàries corresponent a la equació de conservació del moment, més les restriccions algebraiques corresponent a la condició d'incompressibilitat. Mitjançant exemples numèrics es mostra l'aplicabilitat de les metodologies proposades i es comparen la seva eficiència i precisió.This PhD thesis proposes divergence-free Discontinuous Galerkin formulations providing high orders of accuracy for incompressible viscous flows. A new Interior Penalty Discontinuous Galerkin (IPM-DG) formulation is developed, leading to a symmetric and coercive bilinear weak form for the diffusion term, and achieving high-order spatial approximations. It is applied to the solution of the Stokes and Navier-Stokes equations. The velocity approximation space is decomposed in every element into a solenoidal part and an irrotational part. This allows to split the IPM weak form in two uncoupled problems. The first one solves for velocity and hybrid pressure, and the second one allows the evaluation of pressures in the interior of the elements. This results in an important reduction of the total number of degrees of freedom for both velocity and pressure. The introduction of an extra penalty parameter leads to an alternative DG formulation for the computation of solenoidal velocities with no presence of pressure terms. Pressure can then be computed as a post-process of the velocity solution. Other DG formulations, such as the Compact Discontinuous Galerkin method, are contemplated and compared to IPM-DG. High-order Implicit Runge-Kutta methods are then proposed to solve transient incompressible problems, allowing to obtain unconditionally stable schemes with high orders of accuracy in time. For this purpose, the unsteady incompressible Navier-Stokes equations are interpreted as a system of Differential Algebraic Equations, that is, a system of ordinary differential equations corresponding to the conservation of momentum equation, plus algebraic constraints corresponding to the incompressibility condition. Numerical examples demonstrate the applicability of the proposed methodologies and compare their efficiency and accuracy

Polynomial Preserving Recovery For Weak Galerkin Methods And Their Applications

Gradient recovery technique is widely used to reconstruct a better numerical gradient from a finite element solution, for mesh smoothing, a posteriori error estimate and adaptive finite element methods. The PPR technique generates a higher order approximation of the gradient on a patch of mesh elements around each mesh vertex. It can be used for different finite element methods for different problems. This dissertation presents recovery techniques for the weak Galerkin methods and as well as applications of gradient recovery on various of problems, including elliptic problems, interface problems, and Stokes problems. Our first target is to develop a boundary strategy for the current PPR algorithm. The current accuracy of PPR near boundaries is not as good as that in the interior of the domain. It might be even worse than without recovery. Some special treatments are needed to improve the accuracy of PPR on the boundary. In this thesis, we present two boundary recovery strategies to resolve the problem caused by boundaries. Numerical experiments indicate that both of the newly proposed strategies made an improvement to the original PPR. Our second target is to generalize PPR to the weak Galerkin methods. Different from the standard finite element methods, the weak Galerkin methods use a different set of degrees of freedom. Instead of the weak gradient information, we are able to obtain the recovered gradient information for the numerical solution in the generalization of PPR. In the PPR process, we are also able to recover the function value at the nodal points which will produce a global continuous solution instead of piecewise continuous function. Our third target is to apply our proposed strategy and WGPPR to interface problems. We treat an interface as a boundary when performing gradient recovery, and the jump condition on the interface can be well captured by the function recovery process. In addition, adaptive methods based on WGPPR recovery type a posteriori error estimator is proposed and numerically tested in this thesis. Application on the elliptic problem and interface problem validate the effectiveness and robustness of our algorithm. Furthermore, WGPPR has been applied to 3D problem and Stokes problem as well. Superconvergent phenomenon is again observed