Numerical Simulations Of The Quad-Curl Problem

The main purpose of this thesis is to explore efficient numerical methods for the quad-curl problem.Two methods are presented:A curl-conforming weak Galerkin(WG)method and a curl-curl conforming finite element(FE)method.For the first method,we apply curl-conforming Nedelec edge elements and polynomial basis functions defined on element edges to approximate solutions of the quad-curl prob-lem.Based upon the related variational problem,the numerical scheme is constructed by replacing the differential operators with discrete weak differential operators and adding a stabilizer to weakly enforce the continuity of the solution.For polynomial spaces of order? including P?-1 but not P?,error bounds O(h?-1)in the energy norm and O(h?)in the L2 norm are established.Numerical examples validate our theoretical results.For the second method,we construct H2(curl)-conforming finite elements on rectan-gles and triangles,based upon the Nedelec edge elements and the H1 Lagrange elements.Then the conformity and unisolvence of the elements are verified by a straightforward mathematical analysis.For polynomial spaces of order k including P?-1 but not P?,convergence orders O(h?)in the H(curl)norm and O(h?-1)in the H2(curl)norm are established.Numerical experiments confirm our theoretical results.Compared with some existing methods,the curl-conforming WG method is parameter-independent and easy to construct.As for the curl-curl conforming FE method,even the lowest order with a relatively small system can possess high accuracy.