New Weak Galerkin Finite Element Methods For Elliptic And Maxwell Equations

Weak Galerkin(WG)method is a class of non-conforming finite element method,where the usual conforming shape functions and the standard differential operators are replaced by their weak forms,thus,it can be viewed as a natural extension of standard finite element methods(FEMs).The WG method emerges with features of using poly-topal elements intrinsically and constructing high-order elements easily,precisely,it is highly flexible and robust by allowing the use of discontinuous piecewise polynomials and finite element partitions with arbitrary shape of polytopes.WG is also renowned as a parameter-free and absolutely stable method.The theory of WG method has been fur-ther developed as stabilizer-free WG method was introduced,where the stabilizer from the original WG scheme is removed by utilizing larger weak derivative spaces,which simplifies the formulation and reduces the programming complexity arising from the stabilizer.More and more applications of WG method in various scientific computing domain imply that this state-of-art technique is becoming fairly prevalent and signifi-cant.The first subject of this thesis is focused on a variant of WG method(over-penalized weak Galerkin)for Poisson equation,in which double-valued weak functions on inte-rior edges are used for elements(P_k,P_k,[P_k-1]^d)and(P_k,P_k-1,[P_k-1]^d)with spatial dimension d=2,3.We prove that for quasi-uniform triangulations,condition numbers of the stiffness matrices arising from the OPWG method are O(h-^?0^(d)-1-~1),?₀be-ing the penalty exponent and we also introduce a mini-block diagonal preconditioner,which is proven to reduce the condition numbers of stiffness matrices to the magnitude of O(h-~2).We believe that this method can be extended to use polytopal meshes in stabilizer-free regime.Optimal error estimates in a discrete H~1-norm and L~2-norm are established and numerical experiments are presented to demonstrate flexibility,effec-tiveness and reliability of the new method.The following work is focused on interior-penalized weak Galerkin(IPWG)fi-nite element method,in which we use additional terms in OPWG method to avoids ill-conditional linear system.The new method employs element(P_k,P_k,RT _k)and establishes optimal a priori error estimates in discrete H~1-norm and L~2-norm.Some numerical experiments are presented to validate the IPWG method.The third part is to establish an a priori hp error estimate of original weak Galerkin method for second-order elliptic problems.In general,WG method uses the L~2orthog-onal projection of exact solution as an intermediate variable to derive an a priori error estimate,however,this trick seems not to work for hp error estimate on general polyg-onal/polyhedral finite partitions.In this part,we extend polynomial approximation to polygons and introduce an alternative framework for WG error analysis,by which we establish an a priori error estimate explicitly in mesh size h and polynomial degree p.Numerical experiments are presented to demonstrate theoretical results.In the final part,we present an stabilizer-free weak Galerkin method for time-harmonic Maxwell equations.The main idea is to explore more flexibility of the WG discretization of Maxwell equations,and more reliability with relatively lower regu-larity assumption.By following a different regime compare to Mu L.et al 2015,we establish an optimal a priori error estimate for Maxwell equation on general polytopal meshes?moreover,we prove an optimal estimate on simplicial meshes for less regular-ity solution u,×u?H^s+1/2(T_h),s>0.Numerical experiments are presented to validate effectiveness and flexibility of the new scheme.