TY - JOUR
T1 - C-0 elements for generalized indefinite Maxwell equations
AU - Duan,Huoyuan
AU - Lin,Ping
AU - Tan,Roger C. E.
PY - 2012/9
Y1 - 2012/9
N2 - In this paper we develop the C (0) finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H (r) regularity for some r < 1. The ingredients of our method are that two 'mass-lumping' L (2) projectors are applied to curl and div operators in the problem and that C (0) linear element or isoparametric bilinear element enriched with one element-bubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C (0) Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H (r) regularity where r may vary in the interval [0, 1), we obtain the error bound in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds.
AB - In this paper we develop the C (0) finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H (r) regularity for some r < 1. The ingredients of our method are that two 'mass-lumping' L (2) projectors are applied to curl and div operators in the problem and that C (0) linear element or isoparametric bilinear element enriched with one element-bubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C (0) Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H (r) regularity where r may vary in the interval [0, 1), we obtain the error bound in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds.
U2 - 10.1007/s00211-012-0456-x
DO - 10.1007/s00211-012-0456-x
M3 - Article
VL - 122
SP - 61
EP - 99
JO - Numerische Mathematik
T2 - Numerische Mathematik
JF - Numerische Mathematik
SN - 0029-599X
IS - 1
ER -