Error analysis in \(L^ p,1\leq p\leq \infty\), for mixed finite element methods for linear and quasi-linear elliptic problems. (English) Zbl 0698.65060

Durán, Ricardo G.

RAIRO, Modélisation Math. Anal. Numér. 22, No. 3, 371-387 (1988).

Summary: We consider the approximation by mixed finite element methods of second order elliptic problems in \({\mathbb{R}}^ 2\). We show that error estimates in \(L^ p\) follow from stability properties of a weighted \(L^ 2\)- projection on the divergence free vectors of the finite element space. Since we work in two dimensions, we show that this projection is related with a Ritz projection and consequently optimal \(L^ p\) estimates for \(1<p<\infty\) can be derived easily from the known results for the standard finite element method. Also quasi-optimal \(L^{\infty}\) and \(L^ 1\) estimates are obtained. Finally we analyze a quasi-linear problem obtaining similar results than in the linear case.

MSC:

65N15	Error bounds for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

second order elliptic problems; error estimates; stability; Ritz projection; finite element method; quasi-linear

References:

[1]	[1] F. BREZZI, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, R.A.I.R.O., Anal. Numér. 2, 1974, pp. 129-151. Zbl0338.90047 MR365287 · Zbl 0338.90047
[2]	[2] F. BREZZI, J. DOUGLAS Jr., L.D. MARINI, TWOfamilies of mixed finite elements for second order elliptic problems, Numer. Math. 47, 1985, pp. 217-235. Zbl0599.65072 MR799685 · Zbl 0599.65072
[3]	A.P. CALDERON, A. ZYGMUND, On the existence of certain singular integrals, Acta Math. 88, 1952, pp. 85-139. Zbl0047.10201 MR52553 · Zbl 0047.10201
[4]	[4] S. CAMPANATO, G. STAMPACCHIA, Sulle maggiorazioni in L p nella teoria della equazioni ellittiche, Boll. UMI 20, 1965, pp. 393-399. Zbl0142.37604 MR192169 · Zbl 0142.37604
[5]	[5] J. DOUGLAS Jr., R. EWING, M. WHEELER, Approximation of the pressure by a mixed method in the simulation of miscible displacement, R.A.I.R.O., Anal. Numér. 17, 1983, pp. 17-33. Zbl0516.76094 MR695450 · Zbl 0516.76094
[6]	J. DOUGLAS Jr., I. MARTINEZ GAMBA, C. SQUEFF, Simulation of the transient behavior of one dimensional semiconductor device, to appear. Zbl0625.65123 · Zbl 0625.65123
[7]	J. DOUGLAS Jr., J.E. ROBERTS, Mixed finite element methods for second order elliptic problems. Mat. Aplic. Comp. 1, 1982, pp. 91-103. Zbl0482.65057 MR667620 · Zbl 0482.65057
[8]	J. DOUGLAS Jr., J.E. ROBERTS, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44, 1985, pp. 39-52. Zbl0624.65109 MR771029 · Zbl 0624.65109
[9]	[9] M. FORTIN, An analysis of the convergence of mixed finite element methods, R.A.I.R.O., Anal. Numer. 11, 1977, pp. 341-354. Zbl0373.65055 MR464543 · Zbl 0373.65055
[10]	[10] L. GASTALDI, R. H. NOCHETTO, Optimal L \infty -error estimates for nonconforming and mixed finite element methods of lowest order. Numer. Math. 50, 3, 1987, pp. 587-611. Zbl0597.65080 MR880337 · Zbl 0597.65080
[11]	L. GASTALDI, R. H. NOCHETTO, On L \infty -accuracy of mixed finite element methods for second order elliptic problems, to appear. Zbl0677.65103 · Zbl 0677.65103
[12]	L. GASTALDI, R. H. NOCHETTO, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, to appear. Zbl0673.65060 MR1015921 · Zbl 0673.65060
[13]	D. GILBARG, N.S TRUDINGER, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1983. Zbl0562.35001 MR737190 · Zbl 0562.35001
[14]	[14] C. JOHNSON, V. THOMEE, Error estimates for some mixed finite element methods for parabolic type problems, R.A.I.R.O., Anal. Numer. 15, 1981, pp. 41-78. Zbl0476.65074 MR610597 · Zbl 0476.65074
[15]	Y. KWON, F. MILNER, Some new L \infty estimates for mixed finite element methods, to appear. Zbl0624.65098 · Zbl 0624.65098
[16]	Y. KWON, F. MILNER, L \infty -error estimates for mixed methods for semilinear second order elliptic problems, to appear. Zbl0643.65057 · Zbl 0643.65057
[17]	F. MILNER, Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp. 44, 1985, pp. 303-320. Zbl0567.65079 MR777266 · Zbl 0567.65079
[18]	[18] J. NEDELEC, Mixed finite elements in R 3 , Numer. Math. 35, 1980, pp. 315-341. Zbl0419.65069 MR592160 · Zbl 0419.65069
[19]	J. A. NITSCHE, L \infty -convergence of finite element methods, 2nd Conference on Finite Elements, Rennes, France, May 12-14 (1975). MR568857
[20]	R. RANNACHER, R. SCOTT, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38, 1982, pp. 437-445. Zbl0483.65007 MR645661 · Zbl 0483.65007
[21]	P. A. RAVIART, J. M. THOMAS, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method, Lecture Notes in Math N 606, Springer-Verlag, Berlin, 1977, pp. 292-315. Zbl0362.65089 MR483555 · Zbl 0362.65089
[22]	[22] M. SCHECHTER, On L p estimates and regularity, I., Amer. J. Math. 85, 1963, pp. 1-13. Zbl0113.30603 MR188615 · Zbl 0113.30603
[23]	[23] R. SCHOLZ, L \infty -convergence of saddle-point approximations for second order problems, R.A.I.R.O., Anal. Numer. 11, 1977, pp. 209-216. Zbl0356.35026 MR448942 · Zbl 0356.35026
[24]	R. SCHOLZ, Optimal L \infty -estimates for a mixed finite element for elliptic and parabolic problems, Calcolo 20, 1983, pp. 355-377. Zbl0571.65092 MR761790 · Zbl 0571.65092
[25]	R. SCHOLZ, A remark on the rate of convergence for mixed finite element method for second order problems, Numer. Funct. Anal. Optim. 4, 1981-1982, pp. 269-277. Zbl0481.65066 MR665363 · Zbl 0481.65066
[26]	E. STEIN, Singular integrals and differantiability propreties of functions, Princeton University Press, Princeton (1970). Zbl0207.13501 · Zbl 0207.13501