C-0 elements for generalized indefinite Maxwell equations

TY - JOUR

T1 - C-0 elements for generalized indefinite Maxwell equations

A1 - Duan,Huoyuan

A1 - Lin,Ping

A1 - Tan,Roger C. E.

AU - Duan,Huoyuan

AU - Lin,Ping

AU - Tan,Roger C. E.

PY - 2012/9

Y1 - 2012/9

N2 - <p>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 &lt; 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.</p>

AB - <p>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 &lt; 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.</p>

U2 - 10.1007/s00211-012-0456-x

DO - 10.1007/s00211-012-0456-x

M1 - Article

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 1

VL - 122

SP - 61

EP - 99

ER -