Weak Galerkin Finite Element Methods For Partial Differential Equations Of Elliptic Type

The goal of this thesis research is to develop new weak Galerkin (WG) finite element methods for partial differential equations. In particular, the research is fo-cused on three partial differential equations:the biharmonic equation, the steady-state Maxwell’s equations, and a general div-curl problem. The biharmonic equation arises from the theory of thin plate bending. Thin plates refer to those for which the thickness of the plates is much smaller than its lateral dimensions. Maxwell’s equations con-sist of four partial differential equations describing how electrical and magnetic fields are generated and altered by each other and by charges and currents. In Maxwell’s equations, the electrical field and magnetic field are an entirety and undivided, which systematically and integrally summarize the basic laws in electromagnetic field, and predicted the existence of electromagnetic waves. They are named after Scottish mathematician and physicist James Clark Maxwell who published an early form of those equations between1861and1862. The div-curl problem is a basic element for solving Maxwell’s equations. Therefore, designing efficient numerical algorithms for the above mentioned problems is necessary and significant in computational sciences. Our WG method is a new development of the classical Galerkin finite element method, which provides an efficient and robust computational technique for many challenging problems in science and engineering.In Chapter1, we provide some preliminaries which are valuable for all the re-maining presentation in the thesis.In Chapter2, we present a new and efficient numerical algorithm for the bihar-monic equation by using WG finite element methods. The WG finite element scheme is based on a variational formulation of the biharmonic equation that is equivalent to the usual H2-semi norm. Weak Hessian and their approximations, called discrete weak Hessian, are introduced for a class of discontinuous functions defined on a fi-nite element partition of the domain consisting of general polygons or polyhedra. The resulting matrix from the WG finite element method is symmetric, positive definite and parameter free. An error estimate of optimal order is derived in an H2-equivalent norm for the WG finite element solutions. Error estimates in the usual L2norm are established, yielding optimal order of convergence for all the WG finite element al-gorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements).In Chapter3, we present some numerical results to confirm the theory and the accuracy of the WG finite element scheme analyzed in Chapter2. First, the WG fi-nite element scheme is implemented by using the original variational formulation as proposed in Chapter2. The matrix problem arising from the original variational for-mulation contains all the unknowns necessary to represent the functions in the weak finite element space. To reduce the computational cost, we then derive a Schur com-plement form for the WG finite element method in a matrix form. Numerical tests are conducted for the Schur complement problem. The numerical results confirm the mathematical convergence theory developed in Chapter2for the WG finite element method.In Chapter4, we present a hybridized formulation for the WG finite element method for the biharmonic equation. The hybridized weak Galerkin (HWG) scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. An optimal order error estimate is established for the numerical ap-proximations arising from the HWG finite element method. We also derive a compu-tational algorithm, which is equivalent to the Schur complement, by eliminating all the unknown variables in the interior of each element, yielding a significantly reduced system of linear equations for unknowns on the boundary of each element.In Chapter5, we introduce an efficient WG finite element method for the steady-state Maxwell’s equations, where discrete weak curl and discrete weak divergence are used in the corresponding variational formulation, additionally with a stabilization term that enforces a weak continuity of the approximating functions. This WG finite element method is parameter free and is highly flexible and robust for the use of fi- nite element partitions consisting of arbitrary shape-regular polyhedra. Optimal order error estimates are established for the WG finite element approximations in properly-defined discrete norms. An effective implementation of the WG finite element method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only.In Chapter6, we design a class of WG finite element method for general div-curl problems. Based on the Helmholtz decomposition, the div-curl problems are decom-posed into second order elliptic problems plus steady-state Maxwell’s equations. The steady-state Maxwell’s equations here may have different boundary conditions from that in Chapter5. A convergence theory is established for the corresponding WG fi-nite element method by deriving some optimal order error estimates for the numerical approximation.