TY - JOUR

T1 - Primal–dual weak Galerkin finite element methods for elliptic Cauchy problems

AU - Wang, Chunmei

AU - Wang, Junping

N1 - Publisher Copyright:
© 2019 Elsevier Ltd

PY - 2020/2/1

Y1 - 2020/2/1

N2 - The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler–Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal–dual weak Galerkin finite element method. This new primal–dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak L2 topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.

AB - The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler–Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal–dual weak Galerkin finite element method. This new primal–dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak L2 topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.

KW - Elliptic Cauchy problem

KW - Finite element methods

KW - Primal–dual weak Galerkin

UR - http://www.scopus.com/inward/record.url?scp=85071315571&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2019.07.031

DO - 10.1016/j.camwa.2019.07.031

M3 - Article

AN - SCOPUS:85071315571

SN - 0898-1221

VL - 79

SP - 746

EP - 763

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 3

ER -