A STRUCTURE-PRESERVING NONCONFORMING FEM OF NONLINEAR KIRCHHOFF-TYPE EQUATION WITH DAMPING

Introduction: In this thesis we present and analyze the numerical approximation of the second order electromagnetic and acoustic wave equation by the interior penalty (IP) discontinuous Galerkin (DG) finite element method (FEM). In Part I we focus on time-harmonic Maxwell source problems in the high-frequency regime. Part II is devoted to the study of the IP DG FEM for time-dependent acoustic and electromagnetic wave equations. We begin by stating Maxwell's equations in time and frequency domain. We proceed by a variational formulation of Maxwell's equations, and describe the key challenges that are faced in the analysis of the Maxwell operator. Then, we review conforming finite element methods to discretize the second order Maxwell operator. We end this general introduction with some numerical results to highlight the performance and feasibility of conforming FEM for Maxwell's equations. Chapter 2: In this chapter, we introduce and analyze the interior penalty discontinuous Galerkin method for the numerical discretization of the indefinite time-harmonic Maxwell equations in high-frequency regime. Based on suitable duality arguments, we derive a-priori error bounds in the energy norm and the L2-norm. In particular, the error in the energy norm is shown to converge with the optimal order O(hminfs;`g) with respect to the mesh size h, the polynomial degree `, and the regularity exponent s of the analytical solution. Under additional regularity assumptions, the L2-error is shown to converge with the optimal order O(h`+1). The theoretical results are confirmed in a series of numerical experiments on triangular meshes. The thesis' author's principal contributions are the proof of the L2-error bound in Section 2.6, and the proof of Lemma 2.4.1. Chapter 3: We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [44] and can be understood as a non-stabilized variant of the approach proposed in [63]. We show the well-posedness of this approach and derive optimal a-priori error estimates in the energy-norm as well as the L2-norm. The theoretical results are confirmed in a series of numerical experiments. The thesis' author's principal contribution is the proof of the L2-error bound in Section 3.6. Chapter 4: The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second-order scalar wave equation. The resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal; hence, the method is inherently parallel and, leads to fully explicit time integration when coupled with an explicit timestepping scheme. Optimal a priori error bounds are derived in the energy norm and the L2-norm for the semi-discrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order O(hminfs;`g) with respect to the mesh size h, the polynomial degree `, and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the L2-error is shown to converge with the optimal order O(h`+1). Numerical results confirm the expected convergence rates and illustrate the versatility of the method. Chapter 5: We develop the symmetric interior penalty discontinuous Galerkin (DG) method for the spatial discretization in the method of lines approach of the timedependent Maxwell equations in second-order form. We derive optimal a-priori estimates for the semi-discrete error in the energy norm. For smooth solutions, these estimates hold for DG discretizations on general finite element meshes. For low-regularity solutions that have singularities in space, the theoretical estimates hold on conforming, affine meshes. Moreover, on conforming triangular meshes, we derive optimal error estimates in the L2-norm. Finally, we valuate our theoretical results by a series of numerical experiments.

In this work we study finite element methods for two-dimensional Maxwell's equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell's 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's equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell's 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's equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments.

A non-conforming finite volume element method of weighted upstream type for the two-dimensional stationary Navier–Stokes system

In this work we study finite element methods for fourth order variational inequalities. We begin with two model problems that lead to fourth order obstacle problems and a brief survey of finite element methods for these problems. Then we review the fundamental results including Sobolev spaces, existence and uniqueness results of variational inequalities, regularity results for biharmonic problems and fourth order obstacle problems, and finite element methods for the biharmonic problem. In Chapter 2 we also include three types of enriching operators which are useful in the convergence analysis. In Chapter 3 we study finite element methods for the displacement obstacle problem of clamped Kirchhoff plates. A unified convergence analysis is provided for $C^1$ finite element methods, classical nonconforming finite element methods and $C^0$ interior penalty methods. The key ingredient in the error analysis is the introduction of the auxiliary obstacle problem. An optimal $O(h)$ error estimate in the energy norm is obtained for convex domains. We also address the approximations of the coincidence set and the free boundary. In Chapter 4 we study a Morley finite element method and a quadratic $C^0$ interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates with general Dirichlet boundary conditions on general polygonal domains. We prove the magnitudes of the errors in the energy norm and the $L^{\infty}$ norm are $O(h^{\alpha})$, where $\alpha > 1/2$ is determined by the interior angles of the polygonal domain. Numerical results are also presented to illustrate the performance of the methods and verify the theoretical results obtained in Chapter 3 and Chapter 4. In Chapter 5 we consider an elliptic optimal control problem with state constraints. By formulating the problem as a fourth order obstacle problem with the boundary condition of simply supported plates, we study a quadratic $C^0$ interior penalty method and derive the error estimates in the energy norm based on the framework we introduced in Chapter 3. The rate of convergence is derived for both quasi-uniform meshes and graded meshes. Numerical results presented in this chapter confirm our theoretical results.

Parameter dependent finite element analysis for ferronematics solutions

We analyze a new nonconforming Petrov-Galerkin finite element method for solving linear singularly perturbed two-point boundary value problems without turning points. The method is shown to be convergent, uniformly in the perturbation parameter, of orderh 1/2 in a norm slightly stronger than the energy norm. Our proof uses a new abstract convergence theorem for Petrov-Galerkin finite element methods.

A superconvergent point interpolation method (SC‐PIM) is developed for mechanics problems by combining techniques of finite element method (FEM) and linearly conforming point interpolation method (LC‐PIM) using triangular mesh. In the SC‐PIM, point interpolation methods (PIM) are used for shape functions construction; and a strain field with a parameter α is assumed to be a linear combination of compatible stains and smoothed strains from LC‐PIM. We prove theoretically that SC‐PIM has a very nice bound property: the strain energy obtained from the SC‐PIM solution lies in between those from the compatible FEM solution and the LC‐PIM solution when the same mesh is used. We further provide a criterion for SC‐PIM to obtain upper and lower bound solutions. Intensive numerical studies are conducted to verify these theoretical results and show that (1) the upper and lower bound solutions can always be obtained using the present SC‐PIM; (2) there exists an αexact∈(0, 1) at which the SC‐PIM can produce the exact solution in the energy norm; (3) for any α∈(0, 1) the SC‐PIM solution is of superconvergence, and α=0 is an easy way to obtain a very accurate and superconvergent solution in both energy and displacement norms; (4) a procedure is devised to find a αprefer∈(0, 1) that produces a solution very close to the exact solution. Copyright © 2008 John Wiley & Sons, Ltd.

This article investigates the convergence properties of the augmented finite element method (AFEM). The AFEM is here used to model weak discontinuities independently of the underlying mesh. One noticeable advantage of the AFEM over other partition of unity methods is that it does not introduce additional global unknowns. Numerical 2D experiments illustrate the performance of the method and draw comparisons with the finite element method (FEM) and the nonconforming FEM. It is shown that the AFEM converges with an error of in the energy norm. The nonconforming FEM shares the same property while the FEM converges at . Yet, the AFEM is on par with the FEM for certain homogenization problems.

The finite volume element method on the Shishkin mesh for a singularly perturbed reaction–diffusion problem

Locally stabilized [formula omitted]-nonconforming quadrilateral and hexahedral finite element methods for the Stokes equations

In this work, we use the phase velocity method in combination with finite element method to compute the dispersion curve for phase velocity of an ultrasonic pulse traveling in a thin isotropic plate. This method is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized using the finite element method. The variational formulation of the problem corresponding to a fixed value of time is obtained and the existence and uniqueness of the solution is proved. A priori error estimates in the energy norm and in the [Formula: see text] norm are also obtained. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. Numerical experiments show that the velocities computed from the approximated displacements for different frequency values are in good agreement with analytical dispersion curve. This confirms that the proposed symbiosis between finite element and phase velocity method is suitable for computing dispersion curves in more general wave propagation problems, where the geometry is complex and the material is anisotropic.

Advanced Topics in Computational Partial Differential Equations: Numerical Methods and Diffpack Programming

We consider the discretization of the stationary Navier–Stokes/Darcy system in a two-dimensional domain by the non-conforming finite volume element method. We use the standard formulation of the Navier–Stokes/Darcy system in the primitive variables and take as approximation space the non-conforming \(P_{1}\) elements for velocity and piezometric head and piecewise constant elements for the hydrostatic pressure. We prove that the unique solution of the non-conforming finite volume element method converges to the true solution with optimal order for velocity and piezometric head in discrete \(H^{1}\) norm and for pressure in discrete \(L^{2}\) norm, respectively. Finally, some numerical experiments are presented to validate our theoretical results.

Two guaranteed equilibrated error estimators for Harmonic formulations in eddy current problems

This paper is devoted to study the Crouzeix-Raviart (C-R) type nonconforming linear triangular finite element method (FEM) for the nonstationary Navier-Stokes equations on anisotropic meshes. By introducing auxiliary finite element spaces, the error estimates for the velocity in the L2-norm and energy norm, as well as for the pressure in the L2-norm are derived.