Error estimates for least-squares mixed finite elements

A least-squar es mixed finite element method is formulated and applied foi a c lass of second ofdei elhptic problems in two and three dimensionaï domains The pi imaty solution u and the flux a are approximated usmg finite element spaces consisting of piecewise polynomials of de grée k and r respectively The method is nonconforming in the sensé that the boundary condition for the flux approximation cannot be satisfied exactly on the whole boundary F— so it is satisfied only at the nodes on F Optimal Land H-error estimâtes are denved under the standard regulanty assumption on the finite element partition (the UBfè-condition is not requued) The important cases ofk — i and k + l — r are considered

[1]  Jean E. Roberts,et al.  Global estimates for mixed methods for second order elliptic equations , 1985 .

[2]  Max Gunzburger,et al.  On mixed finite element methods for first order elliptic systems , 1981 .

[3]  Pekka Neittaanmäki,et al.  Error estimates for the finite element approximation to a maxwell-type boundary value problem , 1980 .

[4]  Jukka Saranen,et al.  On a inequality of Friedrichs. , 1982 .

[5]  Pekka Neittaanmäki,et al.  On finite element approximation of the gradient for solution of Poisson equation , 1981 .

[6]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[7]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[8]  G. Carey,et al.  Least-squares mixed finite elements for second-order elliptic problems , 1994 .

[9]  J. Oden.  Finite Elements: A Second Course , 1983 .

[10]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[11]  Tsu-Fen Chen,et al.  On least-squares approximations to compressible flow problems , 1986 .

[12]  Pekka Neittaanmäki,et al.  On different finite element methods for approximating the gradient of the solution to the Helmholtz equation , 1984 .

[13]  P. G. Ciarlet,et al.  Interpolation theory over curved elements, with applications to finite element methods , 1972 .

[14]  Graham F. Carey,et al.  Convergence studies of least-squares finite elements for first-order systems , 1989 .

[15]  C. L. Chang,et al.  A least-squares finite element method for the Helmholtz equation , 1990 .

[16]  Über die approximation der lösungen der maxwellschen randwertaufgabe mit der methode der finiten elemente , 1980 .

[17]  H. Saunders,et al.  Finite Elements—A Second Course (Vol. II) , 1987 .

[18]  Finite element approximation of electromagnetic fields in three dimensional space , 1980 .

[19]  Graham F. Carey,et al.  Approximate boundary-flux calculations☆ , 1985 .

[20]  Pekka Neittaanmäki,et al.  On the validity of Friedrichs' inequalities. , 1984 .