A MODIFIED CHARACTERISTIC FINITE ELEMENT METHOD FOR A FULLY NONLINEAR FORMULATION OF THE SEMIGEOSTROPHIC FLOW EQUATIONS

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1 A MODIFIED CHARACTERISTIC FINITE ELEMENT METHOD FOR A FULLY NONLINEAR FORMULATION OF THE SEMIGEOSTROPHIC FLOW EQUATIONS XIAOBING FENG AND MICHAEL NEILAN Abstract Tis paper develops a fully discrete modified caracteristic finite element metod for a coupled system consisting of te fully nonlinear Monge Ampère equation and a transport equation Te system is te Eulerian formulation in te dual space for B J Hoskins semigeostropic flow equations, wic are widely used in meteorology to model frontogenesis To overcome te difficulty caused by te strong nonlinearity, we first formulate at te differential level) a vanising moment approximation of te semigeostropic flow equations, a metodology recently proposed by te autors [17, 18], wic involves approximating te fully nonlinear Monge Ampère equation by a family of fourt-order quasilinear equations We ten construct a fully discrete modified caracteristic finite element metod for te regularized problem It is sown tat under certain mes constraint, te proposed numerical metod converges wit an optimal order rate of convergence In particular, te obtained error bounds sow explicit dependence on te regularization parameter ε Numerical tests are also presented to validate te teoretical results and to gauge te efficiency of te proposed fully discrete modified caracteristic finite element metod Key words semigeostropic flow, fully nonlinear PDE, viscosity solution, modified caracteristic metod, finite element metod, error analysis AMS subject classifications 65M12, 65M15, 65M25, 65M60, 1 Introduction Te semigeostropic flow equations, wic were derived by B J Hoskins [22], are used in meteorology to model slowly varying flows constrained by rotation and stratification Tey can be considered as an approximation of te Euler equations and are tougt to be an efficient model to describe front formation cf [23, 10]) Under certain assumptions and in some appropriately cosen curvilinear coordinates called dual space, see Section 2), tey can be formulated as te following coupled system consisting of te fully nonlinear Monge Ampère equation and te transport equation: 11) 12) 13) 14) detd 2 ψ ) = α in R 3 0, T ], α t + div vα) = 0 in R3 0, T ], αx, 0) = α 0 in R 3 t = 0, ψ Ω, and 15) v = ψ x) = ψ x 2 x 2, x 1 ψ x 1, 0) Here, Ω R 3 is a bounded pysical) domain, α is te density of a probability measure on R 3, and ψ denotes te Legendre transform of a convex function ψ For any w = w 1, w 2, w 3 ), w := w 2, w 1, 0) We note tat none of te variables α, ψ, and v in te system is an original primitive variable appearing in te Euler equations However, all primitive variables can be conveniently recovered from tese non-pysical variables see Section 2 for te details) In tis paper, our goal is to numerically approximate te solution of 11) 15) By inspecting te above system, one easily observes tat tere are tree clear difficulties for acieving te goal First, te equations are posed over an unbounded domain, wic makes numerically solving te system infeasible Second, te ψ -equation is te fully nonlinear THE WORK OF BOTH AUTHORS WAS PARTIALLY SUPPORTED BY THE NSF GRANTS DMS AND DMS Department of Matematics, Te University of Tennessee, Knoxville, TN 37996, USA xfeng@matutkedu) Department of Matematics, Te University of Tennessee, Knoxville, TN 37996, USA neilan@matutkedu) 1

2 2 X FENG AND M NEILAN Monge Ampère equation Numerically, little progress as been made in approximating second-order fully nonlinear PDEs suc as te Monge Ampère equation Tird, equation 14) imposes a nonstandard constraint on te solution ψ, wic is often called a boundary condition of te second kind for ψ in te PDE community cf [3, 10]) As a first step to approximate te solution of te above system, we must solve 11) 13) over a finite domain, U R 3, wic ten calls for te use of artificial boundary condition tecniques For te second difficulty, we recall tat a main obstacle is te fact tat weak solutions called viscosity solutions) for second-order nonlinear PDEs are nonvariational Tis poses a daunting callenge for Galerkin type numerical metods suc as finite element, spectral element, and discontinuous Galerkin metods, wic are all based on variational formulations of PDEs To overcome te above difficulty, recently we introduced a new approac in [17, 18, 19, 20, 25], called te vanising moment metod in order to approximate viscosity solutions of fully nonlinear second-order PDEs Tis approac gives rise to a new notion of weak solutions, called moment solutions, for fully nonlinear second-order PDEs Furtermore, te vanising moment metod is constructive, so practical and convergent numerical metods can be developed based on te approac for computing viscosity solutions of fully nonlinear second-order PDEs Te main idea of te vanising moment metod is to approximate a fully nonlinear second-order PDE by a quasilinear iger order PDE In tis paper, we apply te metodology of te vanising moment metod, and approximate 11) 13) by te following fourt-order quasilinear system: 16) 17) 18) were 19) ε 2 ψ ε + detd 2 ψ ε ) = α ε in U 0, T ], α ε t + div vε α ε ) = 0 in U 0, T ], α ε x, 0) = α 0 x) in R 3 t = 0, v ε := ψ ε x) = ψ ε x 2 x 2, x 1 ψ ε x 1, 0) It is easy to see tat 16) 19) is underdetermined, so extra constraints are required in order to ensure uniqueness To tis end, we impose te following boundary conditions and constraint to te above system: ψ ε = 0 ν on U 0, T ], ψ ε 111) = ε ν on U 0, T ], 112) ψ ε dx = 0 t 0, T ], U were ν denotes te unit outward normal to U We remark tat te coice of 111) intends to reduce te tickness of te boundary layer due to te introduction of te singular perturbation term in 16) see [17] for more discussions) Te boundary condition 110) is used to reduce te reflection due to te introduction of te finite computational domain U It can be regarded as a simple radiation boundary condition An additional consequence of 110) is tat it also effectively overcomes te tird difficulty, wic is caused by te nonstandard constraint 14), for solving system 11) 15) Clearly, 112) is purely a matematical tecnique for selecting a unique function from a class of functions differing from eac oter by an additive constant

3 A modified caracteristic finite element metod for te semigeostropic flows 3 Te specific goal of tis paper is to formulate and analyze a modified caracteristic finite element metod for problem 16) 112) Te proposed metod approximates te elliptic equation for ψ ε by a conforming finite element metod cf [8]) and discretizes te transport equation for α ε by a modified caracteristic metod due to Douglas and Russell [15] We are particularly interested in obtaining error estimates tat sow explicit dependence on ε for te proposed numerical metod Te remainder of tis paper is organized as follows In Section 2, we introduce te semigeostropic flow equations and sow ow tey can be formulated as te Monge Ampère/transport system 11) 15) In Section 3, we apply te metodology of te vanising moment metod to approximate 11) 15) via 16) 112), prove some properties of tis approximation, and also state certain assumptions about tis approximation We ten formulate our modified caracteristic finite element metod to numerically compute te solution of 16) 112) Section 4 mirrors te analysis found in [20] were we analyze te numerical solution of te Monge Ampère equation under small perturbations of te data Section 4 is of independent interests in itself, but te main results will prove to be crucial in te next section In Section 5, under certain mes and time stepping constraints, we establis optimal order error estimates for te proposed modified caracteristic finite element metod Te main idea of te proof is to use te results of Section 4 and an inductive argument In Section 6, we provide numerical tests to validate te teoretical results of te paper Finally, in Section 7 we end wit some conclusions Standard function space notation is adopted in tis paper, we refer to [4, 21, 8] for teir exact definitions In particular,, ) and, denote te L 2 -inner products on U and U, respectively C is used to denote a generic positive constant wic is independent of ε and mes parameters and t 2 Derivation of te Monge Ampère/transport formulation for te semigeostropic flow equations For te reader s convenience and to provide necessary background, we sall first give a concise derivation of te Hoskins semigeostropic flow equations [22] and ten explain ow te Hoskins model is reformulated as a coupled Monge Ampère/transport system Altoug our derivation essentially follows tose of [22, 10, 3], we sall make an effort to streamline te ideas and key steps in a way wic we tougt sould be more accessible to te numerical analysis community Let Ω R 3 denote a bounded domain of te tropospere in te atmospere It is well-known [24] tat if fluids are assumed to be incompressible, teir dynamics in suc a domain Ω is governed by te following incompressible Boussinesq equations, wic are a version of te incompressible Euler equations: 21) Du Dt + p = fu θ ge 3 θ 0 in Ω 0, T ], 22) Dθ = 0 Dt in Ω 0, T ], 23) div u = 0 in Ω 0, T ], 24) u = 0 on Ω 0, T ], were e 3 := 0, 0, 1), u = u 1, u 2, u 3 ) is te velocity field, p is te pressure, θ eiter denotes te temperature in te case of atmospere) or te density in te case of ocean) of te fluid in question θ 0 is a reference value of θ Also D Dt := t + u

4 4 X FENG AND M NEILAN denotes te material derivative Recall tat u := u 2, u 1, 0) Finally, f, assumed to be a positive constant, is known as te Coriolis parameter, and g is te gravitational acceleration constant We note tat te term fu is te so-called Coriolis force, wic is an artifact of te eart s rotation cf [30]) Ignoring te low order) material derivative term in 21) we get 25) 26) were H := H p = fu, p = θ g, x 3 θ 0 ),, 0 x 1 x 2 Equation 25) is known as te geostropic balance, wic describes te balance between te pressure gradient force and te Coriolis force in te orizontal directions Equation 26) is known as te ydrostatic balance in te literature, wic describes te balance between te pressure gradient force and te gravitational force in te vertical direction Define 27) u g := f 1 p) and u ag := u u g, wic are often called te geostropic wind and ageostropic wind, respectively Te geostropic and ydrostatic balances give very simple relations between te pressure field and te velocity field However, te dynamics of te fluids is missing in te description To overcome tis limitation, J B Hoskins [22] proposed te so-called semigeostropic approximation wic is based on replacing te material derivative term Du Dt by Dug in 21) Tis ten leads to te following semigeostropic flow equations in te Dt primitive variables): 28) Du g Dt + Hp = fu in Ω 0, T ], 29) p = θ g x 3 θ 0 in Ω 0, T ], 210) Dθ = 0 Dt in Ω 0, T ], 211) div u = 0 in Ω 0, T ], 212) u = 0 on Ω 0, T ] Te system 28) 212) looks strange, as tere are no explicit dynamic equations for u in te above semigeostropic flow model Also, by te definition of te material derivative, Dug = ug + u )u Dt t g We note tat te full velocity u appears in te last term Sould u be replaced by u g in te material derivative, te resulting model is known as te quasi-geostropic flow equations cf [24]) Due to te peculiar structure of te semigeostropic flow equations, it is difficult to analyze and to numerically solve te equations Te first successful analytical approac as been te one based on te fully nonlinear reformulation 11) 15), wic was first proposed in [5] and was furter developed in [3, 23] see [11] for a different approac) Te main idea of te reformulation is to use time-dependent curved coordinates so te

5 A modified caracteristic finite element metod for te semigeostropic flows 5 resulting system becomes partially decoupled Apparently, te trade-off is te presence of a stronger nonlinearity in te new formulation Te derivation of te fully nonlinear reformulation 11) 15) starts wit introducing te so-called geopotential and geostropic transformation 213) ψ := p f x H 2, Φ := ψ; were x H := x 1, x 2, 0) A direct calculation verifies tat Φ := x H + 1 f 2 Hp θ θ 0 f 2 ge 3 = x H + 1 f u g and consequently, 28) 210) can be rewritten compactly as 214) were DΦ Dt J = = fjφ x), θ θ 0 f 2 ge 3, For any x Ω, let Xx, t) denote te fluid particle trajectory originating from x, ie, Define te composite function dxx, t) = uxx, t), t) t > 0, dt Xx, 0) = x 215) Ψ, t) := Φ, t) X, t) = ΦX, t), t) = ψx, t), t) Ten we ave from 214) 216) Ψx, t) t = fjψx, t) Xx, t)) = fxx, t) Ψx, t)) Since te incompressibility assumption implies tat X is volume preserving, wic is equivalent to 217) Ω gxx, t))dx = det X) = 1, Ω gx)dx g CΩ) We also note tat due to te boundary condition 212), te function x Xx, t) maps Ω into itself To summarize, we ave reduced 28) 211) to 215) 217) It is easy to see tat Ψx, t) is not unique because one as a freedom in coosing te geopotential ψ However, Cullen, Norbury, and Purser [12] also see [10, 3, 23]) discovered te so-called Cullen- Norbury-Purser principle wic says tat Ψx, t) must minimize te geostropic energy at eac time t A consequence of tis minimum energy principle is tat te geopotential

6 6 X FENG AND M NEILAN ψ must be a convex function Using te assumption tat ψ is convex and Brenier s polar factorization teorem [5], Brenier and Benamou [3] proved existence of suc a convex function ψ and a measure-preserving mapping X wic solves 215) 217) To relate 215) 217) wit 11), 12), and 14), let αy, t)dy be te image measure of te Lebesgue measure dx by Ψx, t), tat is gψx, t))dx = gy)αy, t)dy g C c R 3 ) Ω R 3 We note tat te image measure αy, t)dy is te pus-forward Ψ # dx of dx by Ψx, t), and αy, t) is te density of Ψ # dx wit respect to te Lebesgue measure dy Assuming tat ψ is sufficiently regular, it follows from 215) and 217) tat 218) gψx, t))dx = g ψxx, t), t))dx = g ψx, t))dx g C c R 3 ) Ω Ω Using a cange of variable y = ψx, t) on te rigt and te definition of αy, t)dy on te left we get gy)αy, t)dy = R 3 gy) detd 2 ψ y, t))dy R 3 g C c R 3 ), were ψ denotes te Legendre transform of ψ, tat is, Ω 219) Hence ) ψ y, t) = sup x y ψx, t) x Ω αy, t) = detd 2 ψ y, t)), wic yields 11) For convex function ψ, by a property of te Legendre transform we ave ψ y, t) = x Ω Hence ψ Ω, and terefore, 14) olds Finally, for any w Cc [ 1, T ]; R 3 ), it follows from integration by parts and 216) tat Ω wψx, 0), 0) dx = T = 0 Ω T = 0 Ω T 0 Ω wψx, t), t) dwψx, t), t) dt Ψx, t) t dxdt + wψx, t), t) dxdt t wψx, t), t) fxx, t) Ψx, t)) + wψx, t), t) dxdt t Making a cange of variable y = ψx, t) and using te definition of αy, t)dy we get T wy, t) 220) + fvy, t) wy, t) αy, t) dydt + wy, 0)αy, 0) dy = 0, t R 3 0 R 3 were v is as in 15) Hence, αy, t) t + fdiv vy, t)αy, t)) = 0,

7 A modified caracteristic finite element metod for te semigeostropic flows 7 wic gives 13) as f = 1 is assumed in Section 1 We remark tat 218) and 220) are weak formulations of 11) and 12), respectively We also cite te following existence and regularity results for 11)-13) and refer te reader to [3] for teir proofs Teorem 21 Let Ω 0, Ω R 3 be two bounded Lipscitz domain Suppose furter tat α 0 L p R 3 ) wit α 0 0, suppα 0 ) Ω 0, and Ω 0 α 0 x)dx = Ω Ten for any T > 0, p > 1, 11)-13) as a weak solution ψ, α) in te sense of 218) and 220) Furtermore, tere exists an R > 0 suc tat suppαx, t)) B R 0) for all t [0, T ] and α L [0, T ]; L p B R 0))) ψ L [0, T ]; W 1, Ω)) ψ L [0, T ]; W 1, R 3 ) nonnegative, convex in pysical space, convex in dual space Remark 21 a) Te above compact support result for α justifies our approac of solving te original infinite domain problem on a truncated computational domain U, in particular, if U is cosen large enoug so tat B R 0) U b) Since α and ψ are not pysical variables, one needs to recover te pysical variables u and p from α and ψ Tis can be done by te following procedure First, one constructs te geopotential ψ from its Legendre transform ψ Numerically, tis can be done by fast inverse Legendre transform algoritms Second, one recovers te pressure field p from te geopotential ψ using 213) Tird, one obtains te geostropic wind u g and te full velocity field u from te pressure field p using 27) c) Recently, Loeper [23] generalized te above results to te case were α is a global weak probability measure solution of te semigeostropic equations d) As a comparison, we recall tat two-dimensional incompressible Euler equations in te vorticity-stream function formulation) ave te form ω t φ = ω in Ω 0, T ], + div uω) = 0 in Ω 0, T ], u = φ) Clearly, te main difference is tat φ-equation above is a linear equation wile ψ in 11) is a fully nonlinear equation We conclude tis section by remarking tat in te case tat te gravity is omitted, te flow becomes two-dimensional Repeating te derivation of tis section and dropping te tird component of all vectors, we ten obtain a two-dimensional semigeostropic flow model wic as exactly te same form as 11) 15) except tat te definition of te operator ) becomes w := w 2, w 1 ) for w = w 1, w 2 ), and v in 15) is replaced by Similarly, v ε in 19) sould be replaced by v = ψ x 2 x 2, x 1 ψ x 1 ) v ε = ψ ε x 2 x 2, x 1 ψ ε x 1 ) In te rest of tis paper we sall consider numerical approximations of bot two-dimensional and tree-dimensional models 3 Formulation of te numerical metod

8 8 X FENG AND M NEILAN 31 Formulation of te vanising moment approximation As pointed out in Section 1, te primary difficulty for analyzing and numerically approximating te semigeostropic equations 11) 15) is caused by te strong nonlinearity and non-uniqueness of te ψ - equation ie, Monge Ampère equation cf [1, 21]) Te strong nonlinearity makes te equation non-variational, so Galerkin type numerical metods are not directly applicable to te fully nonlinear equation Non-uniqueness is difficult to deal wit at te discrete level because no effective selection criterion is known in te literature wic guarantees picking up te pysical solution ie, te convex solution) Because of te above difficulties, very little progress as been made in te past on developing numerical metods for te Monge Ampère equation and oter fully nonlinear second-order PDEs cf [13, 28, 29]) Very recently, we ave developed a new approac, called te vanising moment metod, for solving te Monge Ampère equation and oter fully nonlinear second-order PDEs cf [17, 18, 19, 20, 25, 26]) Our basic idea is to approximate a fully nonlinear second-order PDE by a singularly perturbed quasilinear fourt-order PDE In te case of te Monge Ampère equation, we approximate te fully nonlinear second-order equation 31) detd 2 w) = ϕ by te following fourt-order quasilinear PDE ε 2 w ε + detd 2 w ε ) = ϕ ε > 0) accompanied by appropriate boundary conditions Te numerics in [18, 19, 20, 25] sows tat for fixed ϕ 0, w ε converges to te unique convex solution w of 31) as ε 0 + A rigorous proof of te convergence in some special cases was carried out in [17] Upon establising te convergence of te vanising moment metod, one can use various well-establised numerical metods suc as finite element, finite difference, spectral and discontinuous Galerkin metods) to solve te perturbed quasilinear fourt-order PDE Remarkably, our experience so far suggest tat te vanising moment metod always converges to te pysical solution Te success motivates us to apply te vanising moment metodology to te semigeostropic model 11) 15), wic leads us to studying problem 16) 112) Remark 31 Since a perturbation term is introduced in 16), it is also natural to introduce a viscosity term ε α on te left-and side of 17) We believe tat tis sould be anoter viable strategy and will furter explore te idea and compare te anticipated new result wit tat of tis paper Since 16) 17) is a quasilinear system, we can define weak solutions for problem 16) 112) in te usual way using integration by parts Definition 31 A pair of functions ψ ε, α ε ) L 0, T ); H 2 U)) L 2 0, T ); H 1 U)) H 1 0, T ); L 2 U)) is called a weak solution to 16) 112) if it satisfies te following integral identities for almost every t 0, T ): 32) 33) 34) 35) ε ψ ε, v ) + detd 2 ψ ε ), v ) = α ε, v) + ε 2, v v H 2 U), α ε ) t, w + v ε α ε, w ) = 0 w H 1 U), α ε, 0), χ ) = α 0, χ ) χ L 2 U), ψ ε, 1) = 0, ere v ε = ψ ε x 2 x 2, x 1 ψ ε x 1, 0) wen d = 3 and v ε = ψ ε x 2 x 2, x 1 ψ ε x 1 ) wen d = 2, and we ave used te fact tat div v ε = 0

9 A modified caracteristic finite element metod for te semigeostropic flows 9 For te rest of te paper, we assume tat tere exists a unique solution to 16) 112) suc tat ψ ε x, t) is convex, α ε x, t) 0, and supp α ε x, t) B R 0) U for all t [0, T ] We also assume ψ ε L 2 0, T ); H s U)) s 3), α ε L 2 0, T ); H p U)) p 2), and tat te following bounds old cf [17]) for almost all t [0, T ]: 36) 37) ψ ε t) H j = Oε 1 j 2 ) j = 1, 2, 3), Φ ε t) L = Oε 1 ), ψ ε t) W j, = Oε 1 j ) j = 1, 2), α ε t) W 1, = Oε 1 ), were Φ ε = cofd 2 ψ ε ) denotes te cofactor matrix of D 2 ψ ε As expected, te proof of te above assumptions is extensive and not easy We do not intend to give a full proof in tis paper However, in te following we sall present a proof of a key assertion, tat is, α ε x, t) 0 in U [0, T ] provided tat α 0 x) 0 in R d d = 2, 3) Clearly, tis assertion is important to ensure tat ψ ε, t) is a convex function for all t [0, T ] Proposition 32 Suppose α ε, ψ ε ) is a regular solution of 16) 112) Assume α 0 x) 0 in R d d = 2, 3), ten α ε x, t) 0 in U [0, T ] Proof For any fixed x, t) U 0, T ], let X ε x, t; s) denote te caracteristic curve passing troug x, t) for te transport equation 17), tat is dx ε x, t; s) = v ε X ε x, t; s), s) s t, ds Xx, t; t) = x Ten te solution α ε at x, t) can be written as α ε x, t) = α 0 X ε x, t; 0)) Hence, α ε x, t) 0 for all x, t) U [0, T ] Te proof is complete 32 Formulation of modified caracteristic finite element metod Let T be a quasiuniform triangulation or rectangular partition of U wit mes size 0, 1) and let V H 2 U) denote a conforming finite element space suc as Argyris, Bell, Bogner Fox Scmit, and Hsie Cloug Tocer finite element spaces [8] wen d = 2) consisting of piecewise polynomial functions of degree r 4) suc tat for any v H s U) s 3) 38) inf v v H j l j v H s, j = 0, 1, 2; l = minr + 1, s v V Also let W be a finite-dimensional subspace of H 1 U) consisting of piecewise polynomials of degree k 1) associated wit te mes T Set 39) 310) V0 := v V ; It is easy to ceck tat v ν = 0 U, V 1 := v V 0 ; v, 1) = 0, W 0 := w W ; w U = 0, τ := 1, vε ) 1 + vε 2 Rd+1 ) τ := τ t, = 1 ) 1 + vε 2 t + vε

10 10 X FENG AND M NEILAN Hence, from 17) we ave 311) α ε τ = vε 2 α ε t + vε α ε ) = 0 Here we ave used te fact tat div v ε = 0 For a fixed positive integer M, let t := T M and t m := m t for m = 0, 1, 2,, M For any x U, let x := x v ε x, t) t It follows from Taylor s formula tat cf [14, 15]) 312) α ε x, t m ) τ = αε x, t m ) α ε x, t m 1 ) t + O t) for m = 1, 2,, M Borrowing te ideas of [14, 15], we propose te following modified caracteristic finite element metod for problem 16) 112): 313) 314) 315) were Algoritm 1: Step 1: Let α 0 be te finite element interpolant or te elliptic projection of α 0 Step 2: For m = 0, 1, 2,, M, find ψ m, αm+1 ) V1 W0 suc tat ε ψ m, v ) + detd 2 ψ m ), v ) = α m, v ) + ε 2, v v V0, ψ m, 1) = 0, ) α m+1 α m, w = 0 w W0, α m := α m x ), x := x v m t, v m := ψ m x) In te case tat W is te finite element space of continuous piecewise linear functions ie, k = 1), we ave te following lemma Lemma 33 Let k = 1 in te definition of W, and suppose tat α 0 0 in Rd d = 2, 3) Ten te solution of Algoritm 1 satisfies α m 0 in U for all m 1 Proof In te case k = 1, 315) immediately implies tat α m+1 P j ) = α m P j ), were P j denote te nodal points of te mes T and P j := P j v m t Suppose tat α mp j) 0 for all j Since te basis functions of te linear element are nonnegative, ten we ave α mp j) 0 for all j Hence, α m+1 P j ) 0 for all j Terefore, te assertion follows from an inductive argument Remark 32 Te positivity of α m for all m 1 gives ope to verify te convexity of ψ m, wic remains an open problem cf [18, 19, 20]) For ig order finite elements ie, k 2), α m migt take negative values for some m > 0 altoug numerical experiments indicate tat te deviation from zero is very small cf Section 6) Let ψ ε, α ε ) be te solution of 16) 112) and ψ m, αm ) be te solution of 313)- 315) In te subsequent sections we prove existence and uniqueness for ψ m, αm ) and provide optimal order error estimates for ψ ε t m ) ψ m and αε t m ) α m under certain mes and time stepping constraints To tis end, we first study 313) independently, wic motivates us to analyze finite element approximations of te Monge Ampère equation wit small perturbations of te data Suc an analysis enables us to bound te error ψ ε t m ) ψ m in terms of te error αε t m ) α m We use similar tecniques to tose developed in [20] to carry out te analysis Wit tis result in and, we use an inductive argument in Section 5 to get te desired error estimates for bot ψ ε t m ) ψ m and αε t m ) α m

11 A modified caracteristic finite element metod for te semigeostropic flows 11 4 Finite element approximations of te Monge Ampère equation wit small perturbations As mentioned above, analyzing te error ψ ε t m ) ψ m motivates us to consider finite element approximations of te following auxiliary problem: for ε > 0, 41) ε 2 u ϕ + detd 2 u ϕ ) = ϕ > 0) in U, 42) u ϕ ν = 0 on U, u ϕ 43) = ε ν on U, 44) u ϕ, 1) = 0, wose weak formulation is defined as seeking u ϕ H 2 Ω) suc tat 45) ε u ϕ, v ) + detd 2 u ϕ ), v ) = ϕ, v ) + ε 2, v 46) u ϕ, 1) = 0 v H 2 U) wit v = 0, ν U We note tat te finite element approximation of a similar Monge Ampère problem was constructed and analyzed in [20], were te Diriclet boundary condition was considered, and te rigt-and side function ϕ is te same in te finite element sceme as in te PDE problem In tis section, we sall study te finite element approximation of 41) 44) in wic ϕ is replaced by ϕ := ϕ + δϕ, were δϕ is some small perturbation of ϕ Specifically, we analyze te following finite element approximation of 41) 44): find u ϕ V 1 suc tat 47) ε u ϕ, v ) + detd 2 u ϕ ), v ) = ϕ, v ) + ε 2, v v V 0 As expected, we sall adapt te same ideas and tecniques as tose of [20] to analyze te above sceme However, we sall omit te details if tey are te same as tose of [20] but igligt te differences if tey are significant, in particular, we sall trace ow te error constants depend on ε and δϕ Also, since te analysis in two dimensions and tree dimensions is essentially te same, we sall only present te detailed analysis in te tree-dimensional case and make comments about te two-dimensional case wen tere is a meaningful difference To analyze sceme 47), we first recall tat cf [20]) te associated bilinear form of te linearization of te operator M ε u ϕ ) := ε 2 u ϕ detd 2 u ϕ ) at te solution u ϕ is given by 48) B[v, w] := ε v, w) + Φ ϕ v, w), were Φ ϕ = cofd 2 u ϕ ) denotes te cofactor matrix of D 2 u ϕ Next, we define a linear operator T ϕ : V1 V1 suc tat for w V1, T ϕ w ) V1 is te solution of following problem: 49) B[w T ϕ w ), v ] = ε w, v ) detd 2 w ), v ) + ϕ, v ) + ε 2, v v V 0 It follows from [20, Teorem 35] tat T ϕ is well-defined Also, it is easy to see tat any fixed point of T ϕ is a solution to 47) We now sow tat if δϕ L 2 is sufficiently

12 12 X FENG AND M NEILAN small, ten indeed, T ϕ as a unique fixed point in a neigborood of u ϕ To tis end, we set B ρ) := v V 1 ; v I u ϕ H 2 ρ, were I u ϕ denotes te finite element interpolant of u ϕ onto V 1 Before we continue, we state a lemma concerning te divergence row property of cofactor matrices A sort proof can be found in [16] Lemma 41 Given a vector-valued function w = w 1, w 2,, w d ) : U R d Assume w [C 2 U)] d Ten te cofactor matrix cof w) of te gradient matrix w of w satisfies te following row divergence-free property: 410) div cof w)) i = d xj cof w)) = 0 for i = 1, 2,, d, j=1 were cof w)) i and cof w)) denote respectively te it row and te i, j)-entry of cof w) Trougout te rest of tis section, we assume u ϕ H s, set l = minr + 1, s, and assume te following bounds compare to tose of [20] and 36)): for j = 1, 2, 3, 411) u ϕ H j = Oε 1 j 2 ), u ϕ W 2, = Oε 1 ), Φ ϕ L = Oε 1 ) We also define te H 2 norm to be te dual norm of te subspace of H 2 U) subject to te omogeneous Neumann boundary condition 42) and te zero mean condition 44) We ten ave te following results Lemma 42 Tere exists a constant C 1 ε) = Oε 1 ) suc tat 412) I u ϕ T ϕ I u ϕ ) H 2 C 1 ε) ε 2 l 2 u ϕ H l + δϕ H 2) Proof To ease notation set s = I u ϕ T ϕ I u ϕ ) and η = I u ϕ u ϕ Ten for any v V 0, we use te mean value teorem to get B[s, v ] = ε I u ϕ ), v ) detd 2 I u ϕ )), v ) + ϕ, v ) + ε 2, v = ε η, v ) + detd 2 u ϕ ) detd 2 I u ϕ )), v ) + δϕ, v ) = ε η, v ) + Υ ε : D 2 u ϕ I u ϕ ), v ) + δϕ, v ), were Υ ε = cofτd 2 I u ϕ ) + 1 τ)d 2 u ϕ ) for τ [0, 1] On noting tat Υ = cofτd 2 I u ϕ ) + 1 τ)d 2 u ϕ ) = det τd 2 I u ϕ ) + 1 τ)d 2 u ϕ ), were D 2 u ϕ denotes te resulting 2 2 matrix after deleting te i t row and j t column of D 2 u ϕ, we obtain Ψ ε ) 2 max τd 2 I u ϕ ) kl + 1 τ)d 2 u ϕ ) kl ) 2 k i,l j C max k i,l j D2 u ϕ ) kl 2 C D 2 u ϕ 2 L

13 A modified caracteristic finite element metod for te semigeostropic flows 13 Hence, from 411) it follows tat Υ ε L = Oε 2 ) Tus, B[s, v ] ε η L 2 v L 2 + Cε 2 D 2 η L 2 v L 2 + δϕ H 2 v H 2 C ε 2 η H 2 + δϕ H 2) v H 2 Finally, using te coercivity of B[, ] we get s H 2 C 1 ε) ε 2 l 2 u ϕ H l + δϕ H 2) Te proof is complete Lemma 43 Tere exists 0 > 0 suc tat for 0, tere exists a ρ = ρ, ε) suc tat for any v, w B ρ) tere olds 413) T ϕ v ) T ϕ w ) H v w H 2 Proof From te definitions of T ϕ v ) and T ϕ w ) we get for any z V 0 B[T ϕ v ) T ϕ w ), z ] = Φ ϕ v w ), z ) + detd 2 v ) detd 2 w ), z ) Adding and subtracting detd 2 v µ ) and detd2 w µ ), were vµ and wµ denote te standard mollifications of v and w, respectively, yields B[T ϕ v ) T ϕ w ), z ] = Φ ϕ v w ), z ) + detd 2 v µ ) detd2 w µ ), z ) + detd 2 v ) detd 2 v µ ), z ) + detd 2 w µ ) detd2 w ), z ) = Φ ϕ v w ), z ) + Ψ : D 2 v µ D2 w µ ), z ) + detd 2 v ) detd 2 v µ ), z ) + detd 2 w µ ) detd2 w ), z ), were Ψ = cofd 2 v µ + τd2 w µ D2 v µ )) for τ [0, 1] Using Lemma 41 and Sobolev s inequality we ave 414) B[T ϕ v ) T ϕ w ), z ] = Φ ϕ Ψ ) v w ), z ) + Ψ v v µ ), z ) + Ψ w µ w ), z ) + detd 2 v ) detd 2 v µ ), z ) + detd 2 w µ ) detd2 w ), z ) C Φ ϕ Ψ L 2 v w H 2 + Ψ L 2[ v v µ H 2 + w w µ ] H 2 + detd 2 v ) detd 2 v µ ) L 2 + detd 2 w ) detd 2 w µ ) L z 2 H 2 It follows from te mean value teorem tat Φ ϕ Ψ ) L 2 = detd 2 u ϕ ) detd 2 v µ + τd 2 w µ D 2 v µ )) L 2 = Λ : D 2 u ϕ D 2 v µ + τd 2 w µ D 2 v µ ))) L 2, were Λ = cofd 2 u ϕ + λd 2 v µ + τd 2 w µ D 2 v µ ))) R 2 2 for λ [0, 1]

14 14 X FENG AND M NEILAN We bound Λ L as follows: Λ L = cofd 2 u ϕ + λd 2 v µ + τd 2 w µ D 2 v µ ))) L = D 2 u ϕ + λd 2 v µ + τd 2 w µ D 2 v µ )) L C ε ρ + D 2 v µ D2 v L + D 2 w µ D2 w L ), were we used te triangle inequality followed by te inverse inequality and 411) Combining te above two inequalities we get Φ ϕ Ψ ) L 2 Λ L D 2 u ϕ D 2 v µ + τd 2 w µ D 2 v µ )) L 2 C ε ρ + D 2 v µ D2 v L + D 2 w µ ) D2 w L l 2 u ϕ H l + ρ + D 2 v D 2 v µ L 2 + D2 w µ D2 w L 2) Hence, 415) Φ ϕ Ψ L 2 C ε ρ + D 2 v µ D2 v L + D 2 w µ ) D2 w L l 2 u ϕ H l + ρ + D 2 v D 2 v µ L 2 + D2 w µ D2 w L 2) Applying 415) to 414) and letting µ 0 yields B[T ϕ v ) T ϕ w ), z ] C ε ρ ) l 2 u ϕ H l + ρ ) v w H 2 z H 2 Using te coercivity of B[, ] we get 416) T ϕ v ) T ϕ w ) H 2 Cε 1 ε ρ ) l 2 u ϕ H l + ρ ) v w H 2 Finally, setting 0 = O from 416) tat ε 2 u ϕ H l ) 1 l 2, 0, and ρ = Ominε 2, ε 3 2 ), it ten follows T ϕ v ) T ϕ w ) H v w H 2 v, w B ρ) Te proof is complete Wit te elp of te above two lemmas, we are ready to state and prove te main results of tis section Teorem 41 Suppose δϕ H 2 = Ominε 3, ε ) Ten tere exists an 1 > 0 suc tat for 1, tere exists a unique solution u ϕ V 1 solving 47) Furtermore, tere olds te following error estimate: u ϕ u ϕ H 2 C 2ε) ) 417) ε 2 l 2 u ϕ H l + δϕ H 2 wit C 2 ε) = Oε 1 ) Proof To sow te first claim, we set ε 4 1 = O min u ϕ H l ) 2 2l 7, ε 5 u ϕ H l ) 1 ) l 2 Fix 1 and set ρ 1 = 2C 1 ε) ε 2 l 2 u ϕ H l + δϕ H 2) Ten we ave ρ1 Cminε 2, ε 3 2

15 A modified caracteristic finite element metod for te semigeostropic flows 15 Next, let v B ρ 1 ) Using te triangle inequality and Lemmas 42 and 43 we get I u ϕ T ϕ v ) H 2 I u ϕ T ϕ I u ϕ ) H 2 + T ϕ I u ϕ ) T ϕ v ) H 2 C 1 ε) ) ε 2 l 2 u ϕ 1 H l + δϕ H I u ϕ v H 2 ρ ρ 1 2 = ρ 1 Hence, T ϕ v ) B ρ 1 ) In addition, by 413) we know tat T ϕ is a contraction mapping in B ρ 1 ) Tus, Brouwer s Fixed Point Teorem [21] guarantees tat tere exists a unique fixed point u ϕ B ρ 1 ), wic is a solution to 47) Finally, using te triangle inequality we get u ϕ u ϕ H 2 uϕ I u ϕ H 2 + I u ϕ u ϕ H 2 Cl 2 u ϕ H l + ρ 1 C 2 ε) ε 2 l 2 u ϕ H l + δϕ H 2) Teorem 42 In addition to te ypoteses of Teorem 41, assume tat te linearization of M ε at u ϕ see 48)) is H 3 -regular wit te regularity constant C s ε) Furtermore, assume tat δϕ H 2 = OCs 1 ε)ε 2 ) Ten tere exists an 2 > 0 suc tat for 2, tere olds 418) u ϕ u ϕ H 1 C sε) C 3 ε) l 1 u ϕ H l + C 4 ε) + 1) δϕ H 2, were C 3 ε) = C 2 ε)ε 5 2 and C 4 ε) = C 2 ε)ε 1 2 Proof Let e ϕ := u ϕ u ϕ and uϕ,µ denote a standard mollification of u ϕ We note tat e ϕ satisfies te following error equation: 419) ε e ϕ, z ) + detd 2 u ϕ ) detd2 u ϕ ), z ) + δϕ, z ) = 0 z V 0 Using 419), te mean value teorem, and Lemma 41 we ave 420) 0 = ε e ϕ, z ) Φ u ϕ,µ u ϕ ), z ) + δϕ, z ) + detd 2 u ϕ ) detd2 u ϕ,µ ), z ), were Φ = cofd 2 u ϕ,µ + τd 2 u ϕ D 2 u ϕ,µ )) for τ [0, 1] Next, let φ V 0 H 3 be te unique solution to te following problem: Te regularity assumption implies tat B[φ, z] = e ϕ, z) z V 0 421) φ H 3 C s ε) e ϕ L 2

16 16 X FENG AND M NEILAN We ten ave 422) e ϕ 2 L 2 = ε eϕ, φ) + Φ ϕ φ, e ϕ ) = ε e ϕ, φ I φ)) + Φ ϕ e ϕ, φ I φ)) + ε e ϕ, I φ)) + Φ ϕ e ϕ, I φ)) ε e ϕ, I φ)) Φ u ϕ u ϕ,µ ), I φ)) detd 2 u ϕ ) detd2 u ϕ,µ ), I φ) δϕ, I φ) = ε e ϕ, φ I φ)) + Φ ϕ e ϕ, φ I φ)) + Φ ϕ Φ) e ϕ, I φ)) + Φ u ϕ,µ u ϕ ), I φ)) + detd 2 u ϕ,µ ) detd2 u ϕ ), I φ) δϕ, I φ) ε e ϕ L 2 φ I φ) L 2 + C Φ ϕ L 2 e ϕ H 2 φ I φ H 2 + C Φ ϕ Φ L 2 e ϕ L 2 I φ) L + C Φ L 2 u ϕ,µ u ϕ H 2 I φ H 2 + detd 2 u ϕ ) detd2 u ϕ,µ ) L 2 I φ L 2 + δϕ H 2 I φ H 2 ε 1 2 e ϕ H 2 + Φ ϕ Φ L 2 e ϕ L 2 + Φ L 2 u ϕ,µ C + detd 2 u ϕ ) detd2 u ϕ,µ ) L 2 + δϕ H 2 φ H 3 u ϕ L 2 We bound Φ ϕ Φ L 2 as follows: Φ ϕ Φ) L 2 = cofd 2 u ϕ ) cofd 2 u ϕ,µ + τd 2 u ϕ D 2 u ϕ,µ )) ) L 2 = detd 2 u ϕ ) detd 2 u ϕ,µ + τd 2 u ϕ D 2 u ϕ,µ )) L 2 = Λ : D 2 u ϕ D 2 u ϕ,µ + τd 2 u ϕ D 2 u ϕ,µ ))) L 2 2 Λ L D 2 u ϕ D 2 u ϕ L 2 + D2 u ϕ D2 u ϕ,µ ) L 2, were Λ = cofd 2 u ϕ + λd 2 u ϕ,µ + τd 2 u ϕ D 2 u ϕ,µ ))), λ [0, 1] Notice tat we ave abused te notation Λ by defining it differently tan in te proof of Lemma 43 To estimate Λ L, we note tat Λ R 2 2 Tus for 1 Λ L C D 2 u ϕ L + D 2 u ϕ D 2 u ϕ L + D2 u ϕ D2 u ϕ,µ ) L C ε 1 + C 2 ε) ε 2 l 7 2 u ϕ H l δϕ H 2 C ε 1 + D 2 u ϕ,µ D 2 u ϕ L ), ) + D 2 u ϕ,µ D 2 u ϕ L ) were we ave used te triangle inequality, te inverse inequality, and 411) Terefore, Φ ϕ Φ ) 423) ε L 2 C ε 1 + D 2 u ϕ,µ D 2 u ϕ L C 2 ε) ) ε 2 u ϕ H l + δϕ H 2 + D 2 u ϕ D2 u ϕ,µ L ) 2 Using 423) and setting µ 0 in 422) yields e ϕ 2 L 2 C ε 1 2 e ϕ H 2 + δϕ H 2 + ε 1 C 2 ε) ε 2 l 2 u ϕ H l + δϕ H 2) e ϕ L 2 φ H 3

17 A modified caracteristic finite element metod for te semigeostropic flows 17 It follows from 421) tat e ϕ L 2 C s ε) ε 1 2 e ϕ H 2 + δϕ L 2 Set 2 = O ε 4 C sε) u ϕ H l + ε 1 C 2 ε) ε 2 l 2 u ϕ H l + δϕ H 2) e ϕ L 2 ) 1 l 2 On noting tat δϕ H 2 εc s ε)c 2 ε)) 1 we ave for min 1, 2 e ϕ L 2 C s ε) C 2 ε)ε 5 2 l 1 u ϕ H l + ε 1 2 C2 ε) + 1 ) δϕ H 2 Tus, 418) follows from Poincare s inequality Te proof is complete Remark 41 Let ψ m, αm ) be generated by Algoritm 1 If αε t m ) α m L 2 = O minε 3, ε 2 3 2, Cs 1 ε)ε 2 ), ten by Teorems 41 and 42, for min 1, 2, tere exists a unique ψ m V 1 solving 313), were ) ε 4 2 2l 7, 1 = O min ψ ε L 2 [0,T ];H l ) ) ε 4 1 l 2 2 = C s ε) ψ ε L 2 [0,T ];H l ) Furtermore, we ave te following error bounds: ε 5 ψ ε L 2 [0,T ];H l ) ) 1 ) l 2, ψ ε t m ) ψ m H 2 C 2 ε) 424) ε 2 l 2 ψ ε t m ) H l + α ε t m ) α m L 2), 425) ψ ε t m ) ψ m H 1 C s ε) C 3 ε) l 1 ψ ε t m ) H l ) + C 4 ε) + 1) α ε t m ) α m L 2) Remark 42 In te two dimensional case, ) ε 5 1 l 2 ) 2 ε 3 1 l 2 1 = O, ψ ε 2 = O, L 2 [0,T ];H l ) C s ε) ψ ε L 2 [0,T ];H l ) ψ ε t m ) ψ m H 2 C 2 ε) ε 1 2 l 2 ψ ε t m ) H l + α ε t m ) α m L 2), and 425) olds wit C 3 ε) = C 2 ε)ε 1 2 and C 4 ε) = C 2 ε)ε 1 2 Furtermore, we only require α ε t m ) α m L 2 = Ominε2, Cs 1 ε)ε) 5 Error analysis for Algoritm 1 In tis section we sall derive error estimates for te solution of Algoritm 1 Tis will be done by using an inductive argument based on te error estimates of te previous section Before stating te first main result of tis section, we cite some well-known error estimate results for te elliptic projection of αt m ), wic we denote by χ m W 0 Let ω m ) := α, t m ) χ m ), ten te following estimates old for αt m ) χ m, m 1 cf [4, 8]): 51) ω L 2 [0,T ];L 2 ) + ω L 2 [0,T ];H 1 ) C j α L 2 [0,T ];H j ), ω t L 2 [0,T ];L 2 ) + ω t L 2 [0,T ];H 1 ) C j α t L 2 [0,T ];H j ), ω m W 1, C j 1 αt m ) W j,,

18 18 X FENG AND M NEILAN were j := mink + 1, p As in Section 4, we set l = minr + 1, s Teorem 51 Tere exists 3 > 0 suc tat for min 1, 2, 3 tere exists t 1 > 0 suc tat for t min t 1, 2 52) max 0 m M αε t m ) α m L 2 C 5 ε) t αττ ε L 2 [0,T ] R 3 ) + j[ ] α ε L 2 [0,T ];H j ) + αt ε L 2 [0,T ];H j ) + C 6 ε) l ψ ε L 2 [0,T ];H l ), 53) max 0 m M ψε t m ) ψ m H 2 C 7 ε) t αττ ε L 2 [0,T ] R 3 ) + j[ ] α ε L 2 [0,T ];H j ) + αt ε L 2 [0,T ];H j ) + C 6 ε) l 2 ψ ε L 2 [0,T ];H l ), 54) max 0 m M ψε t m ) ψ m H 1 C 8 ε) t αττ ε L 2 [0,T ] R 3 ) + j[ ] α ε L 2 [0,T ];H j ) + αt ε L 2 [0,T ];H j ) + C 6 ε) l 1 ψ ε L 2 [0,T ];H l ), were C 5 ε) = Oε 1 ), C 6 ε) = C s ε)c 3 ε), C 7 ε) = C 2 ε)c 5 ε), C 8 ε) = C s ε)c 5 ε), and C s ε) is defined in Teorem 42 Proof We break te proof into five steps Step 1: Te proof is based on two inductive ypoteses, were we assume for m = 0, 1,, k, 55) 56) 4 = O α ε t m ) α m L 2 = O minε 3, ε 2 3 2, C 1 s ε)ε 2 ), D 2 ψ m L = Oε 1 ) We first sow tat te claims of te teorem old wen k = 0 Let ) 1 ) 2 ε 3 j ε 2 2j 3,, min From 51) we ave for 4 α 0 H j α 0 H j ε 2 C s ε) α 0 H j ) 1 j Cminε 3, ε 2 3 2, C 1 s ε)ε 2 α 0 α 0 L 2 C j α 0 H j By Remark 41, tere exists ψ 0 solving 313) On noting tat 1 C we ave for min 1, 2, 4 ) ε 2 ψ ε 0) H l D 2 ψ 0 L D 2 ψ ε 0) L D 2 ψ ε 0) D 2 ψ 0 L 2 C ε C2 ε) ε 2 l 2 ψ ε 0) H l + j α0 ε H j) ) Cε 1 ) 2 2l 7, Te remaining four steps are devoted to sowing tat te estimates old for m = k+1 Step 2: Let ξ m := α m χm By 315) and 17), and a direct calculation we get 57) ξ m+1 ξ m, ξ m+1) = t ατt ε m+1 ) α ε t m+1 ) α ε t m )), ξ m+1) + ω m+1 ω m, ξ m+1),

19 A modified caracteristic finite element metod for te semigeostropic flows 19 were ξ m := ξ m x ), α ε t m) := α ε x, t m ), and ω m := ωm x ) We now estimate te rigt-and side of 57) To bound te first term, we write t α ε τx, t m+1 ) [ α ε x, t m+1 ) α ε x, t m ) ] Using te identity and 36) we obtain = t α ε τx, t m+1 ) [ α ε x, t m+1 ) α ε x, t m ) ] + [ α ε x, t m ) α ε x, t m ) ] t α ε τx, t m+1 ) [ α ε x, t m+1 ) α ε x, t m ) ] = x,tm+1 ) x,t m) xτ) x 2 + tτ) t m ) 2 α ε ττdτ 58) t ατt ε m+1 ) [ α ε t m+1 ) α ε t m ) ] 2 L 2 x,tm+1 ) = xτ) x 2 + tτ) t m ) 2 αττdτ ε 2 dx R 3 x,t m) t vε t m+1 ) R 3 C t 2 v ε t m+1 ) L R 3 C t 2 α ε ττ 2 L 2 [t m,t m+1 ] R 3 ), x,tm+1 ) x,t m) x,tm+1 ) x,t m) αττdτ ε 2 dx α ε ττ 2 dτdx were α ε t m ) := α ε x, t m ) Since ten 59) α ε x, t m ) α ε x, t m ) = α ε t m ) α ε t m ) 2 L 2 = t 2 R α ε x + s x x ), t m ) x x )ds, α ε x + s x x ), t m ) v m v ε t m ))ds 2 dx t 2 α ε t m ) 2 W 1, vm v ε t m ) 2 L 2 Cε 2 t 2 v m v ε t m ) 2 L 2 Using 58) 59), we can bound te first term on te rigt-and side of 57) as follows: t α ε τ t m+1 ) [ α ε t m+1 ) α ε t m ) ], ξ m+1) 510) C t 2 α ττ 2 L 2 [t m,t m+1 ] R 3 ) + ε 2 v m v ε t m ) 2 L 2 ) ξm+1 2 L 2 To bound te second term on te rigt-and side of 57), writing ω m+1 x) ω m x ) = ω m+1 x) ω m x) ) + ω m x) ω m x) ) + ω m x) ω m x ) ),

20 20 X FENG AND M NEILAN we ten ave 511) ω m+1 ω m 2 L 2 t ω t 2 L 2 [t m,t m+1 ] R 3 ) Next, it follows from ω m x) ω m x) = t tat set ω m := ω m x)) 1 0 ω m x + s x x)) v ε t m )ds 512) ω m ω m 2 L 2 C t2 v ε t m ) 2 L ωm 2 H 1 C t2 ω m 2 H 1 Finally, using te identity we get ω m x) ω m x ) = t 1 0 ω m x + s x x)) v ε t m ) v m )ds, ω m ω m 2 L 2 C t2 ω m 2 W 1, vε t m ) v m 2 L 2 513) C t 2 v ε t m ) v m 2 L 2 Combining 511) 513), we ten bound te second term on te rigt-and side of 57) as follows: ω m+1 ω m, ξ m+1) C 514) t ω t 2 L 2 [t m,t m+1 ] R 3 ) + t2 ω m 2 H 1 + t 2 v ε t m ) v m 2 L 2 ) ξm+1 2 L 2 Step 3: To get a lower bound of ξ m+1 ξ m, ξ m+1 ), let F m x) := x tv m x) We ten ave detj Fm ) = 1 + t ψ m x 1 x 1 ψ m x 2 x 2 ψ m x 1 x 2 ) 2 ψ m x 1 x 1 + ψ m x 2 x 2 ) ), were J Fm denotes te Jacobian of F m, and we ave omitted te subscript for notational convenience Letting t 0 = Oε), we can conclude from te inductive ypoteses tat for t t 0, F m is invertible and detj F 1 ) = 1 + Cε 2 t 2 From tis result we get m 515) Tus, 516) ξ m+1 ξ m, ξ m+1 ) 1 2 ξ m 2 L 2 = 1 + ε 2 t 2 ) ξ m 2 L 2 [ ξ m+1, ξ m+1 ) ξ m, ξ m ) ] = 1 2 ξ m+1 2 L ε 2 t 2 ) ξ m 2 L 2 ) Step 4: Combining 57), 510), 514), 516), and using te inductive ypoteses

21 and Remark 41 yields A modified caracteristic finite element metod for te semigeostropic flows 21 ξ m+1 2 L 2 ξm 2 L 2 Cε 2 t 2 α ε ττ 2 L 2 [t m,t m+1 ] R 3 ) + ωm 2 H 1 + vm v ε t m ) 2 L 2 ) + t ω t 2 L 2 [t m,t m+1 ] R 3 ) + t2 ξ m 2 L 2 Cε 2 t 2 α ε ττ 2 L 2 [t m,t m+1 ] R 3 ) + ωm 2 H 1 + C 2 s ε) C 2 3ε) 2l 2 ψ ε t m ) 2 H l + C 2 4ε) 2 + 1) α ε t m ) α m 2 H 2 ) ) + t ω t 2 L 2 [t m,t m+1 ] R 3 ) + t2 ξ m 2 L 2 It follows from te inequality α ε t m ) α m H 2 α ε t m ) α m L 2 ξ m L 2 + ω m L 2 tat ξ m+1 2 L 2 ξm 2 L 2 Cε 2 t 2 α ε ττ 2 L 2 [t m,t m+1 ] R 3 ) + C2 4ε) 2 + 1) ω m 2 H 1 + C 2 s ε)c 2 3ε) 2l 2 ψ ε t m ) 2 H l ) + t ω t 2 L 2 [t m,t m+1 ] R 3 ) + C 2 4ε) 2 + 1) t 2 ξ m 2 L 2 Applying te summation operator k m=0 and noting tat ξ0 = 0 we get ξ k+1 2 L 2 Cε 2 t 2 α ε ττ 2 L 2 [0,T ] R 3 ) + t [C 2 4ε) 2 + 1) ω 2 L 2 [0,T ];H 1 ) + C 2 s ε)c 2 3ε) 2l 2 ψ ε 2 L 2 [0,T ];H l ) + ω t 2 L 2 [0,T ] R 3 ) + t 2 C 2 4ε) 2 + 1) k ξ m 2 L, 2 m=0 wic by an application of te discrete Gronwall s inequality yields ] 517) ξ k+1 L 2 Cε ε 1 C 4 ε) + 1) t ) k+1 t αττ ε L 2 [0,T ] R 3 ) + [ t C 4 ε) + 1) ω L 2 [0,T ];H 1 ) + C s ε)c 3 ε) l 1 ψ ε L 2 [0,T ];H l ) + ω t L 2 [0,T ] R 3 ) ] We note 1 = OC4 1 ε)) = Oε 3 2 ) Tus, for min 1, 2, 4 and t min t 0, 2, we ave from 517), te triangle inequality, and 51) tat 518) α ε t k+1 ) α k+1 L 2 C 5 ε) t αττ ε L 2 [0,T ] R 3 ) + j[ α ε L 2 [0,T ];H j ) + αt ε L 2 [0,T ];H )] j + C6 ε) l ψ ε L 2 [0,T ];H l )

22 22 X FENG AND M NEILAN Tus, by Remark 41, we obtain te following estimates: 519) ψ ε t k+1 ) ψ k+1 H 2 C 7 ε) t αττ ε L 2 [0,T ] R 3 ) + j[ α ε L 2 [0,T ];H j ) + αt ε L 2 [0,T ];H )] j + C6 ε) l 2 ψ ε L 2 [0,T ];H l ), 520) ψ ε t k+1 ) ψ k+1 H 1 C 8 ε) t αττ ε L 2 [0,T ] R 3 ) + j[ α ε L 2 [0,T ];H j ) + αt ε L 2 [0,T ];H )] j + C6 ε) l 1 ψ ε L 2 [0,T ];H l ) and let Step 5: We now verify te induction ypoteses Set C 9 ε) = ε 2 minε, Cs 1 ε), C 10 ε) = C 5 ε) α ε L 2 [0,T ];H j ) + αt ε L 2 [0,T ];H j )), C 11 ε) = C 5 ε)c 6 ε) ψ ε L 2 [0,T ];H l ), C 9 ε) 5 = O min C 11 ε) t 1 = O ) 1 l, ε 2 minc 9 ε), ε C 5 ε) α ε ττ L 2 [0,T ] R 3 ) C 11 ε) ) ) 2 ) 1 ) 2 2l 3 C 9 ε) j ε 2 2j 3 ),, C 10 ε) C 10 ε) On noting tat t 1 t 0, it follows from 518) tat for min 1, 2, 4, 5 and t min t 1, 2 α ε t k+1 ) α k+1 L 2 Cminε 3, ε 2 3 2, C 1 s ε) Tus, te first inductive ypotesis 55) olds Finally, let ) 2 ε 2l 7 6 = O C 6 ε)c 7 ε) ψ ε L 2 [0,T ];H l ) By te definitions of 5, 7 and t 1, 519), 36), and te inverse inequality we ave for min 1, 2, 4, 5, 6 and t min t 1, 2 D 2 ψ k+1 L D 2 ψ ε t k+1 ) L + C 3 2 D 2 ψ ε t k+1 ) D 2 ψ k+1 Cε C7 ε) t αττ ε L 2 [0,T ] R 3 ) L 2 + j[ α ε L 2 [0,T ];H j ) + α ε t L 2 [0,T ];H j )] + C6 ε) l 2 ψ ε L 2 [0,T ];H l ) Cε 1 Terefore, te second inductive ypotesis 56) olds, and te proof is complete by setting 3 = min 1, 2, 4, 5, 6 Remark 51 In te two dimensional case, t 1 = O minε 2, Cs 1 ε)ε C 5 ε) αττ ε L 2 [0,T ] R 2 ) Remark 52 Recalling te definitions of V and W, we require k r 2 in order to obtain optimal order error estimate for ψ m in te H2 -norm )

23 A modified caracteristic finite element metod for te semigeostropic flows 23 6 Numerical experiments In tis section we sall present tree two-dimensional numerical experiments Te first two experiments are done in te domain U = 0, 1) 2, wile te tird experiment uses U = 0, 6) 2 In all tree experiments te fift degree Argyris plate finite element cf [8]) is used to form V, and te cubic Lagrange element is employed to form W We recall tat see Section 2) te two-dimensional geostropic flow model as exactly te same form as 11) 15) except v and v ε in 15) and 19) are replaced respectively by v = ψ x 2 x 2, x 1 ψ x 1 ), v ε = ψ ε x 2 x 2, x 1 ψ ε x 1 ) 61 Test 1 Te purpose of tis test is twofold First, we compute α m and ψm to view certain properties of tese two functions Specifically, we want to verify α m > 0 and tat ψ m is strictly convex for m = 0, 1,, M Second, we calculate ψ ψ ε and α αε for fixed = 0023 and t = in order to approximate ψ ψ ε and α α ε We set to solve problem 313) 315) wit te rigt-and side of 315) being replaced by F, w ), and V1 and W0 being replaced by Vg N and Wg D, respectively, were Vg N t) := v V v ; = g N, v, 1) = ct), ct) := ψ, 1), ν U Wg D t) := w W ; w U = g D We use te following test functions and parameters g N x, t) = te tx2 1 +x2 2 )/2 x 1 ν x1 + x 2 ν x2 ), g D x, t) = t tx x 2 2))e tx2 1 +x2 2 ), F x, t) = t 2 + 4tx x 2 2) + t 2 x x 2 2) 2) e tx2 1 +x2 2 ), so tat te exact solution of 11) 15) is given by ψ x, t) = e tx2 1 +x2 2 )/2, αx, t) = t tx x 2 2))e tx2 1 +x2 2 ) We record te computed solutions and plot te errors against ε in Figure 61 at t m = 025 Te figure sows tat ψ t m ) ψ m H 2 = Oε 1 4 ), and since we ave set bot and t very small, tese results suggest tat ψ t m ) ψ ε t m ) H 2 = Oε 1 4 ) Similarly, we argue ψ t m ) ψ ε t m ) H 1 = Oε 3 4 ) and ψ t m ) ψ ε t m ) L 2 = Oε) based on our results We note tat tese are te same convergence results as tose found in [18, 19, 20, 25], were te single Monge Ampère equation was considered We also notice tat tis test suggests tat αt m ) α ε t m ) L 2 may not converge in te limit of ε 0, wic suggests tat convergence can only be possible in a weaker norm suc as H 2 Next, we plot α m, and ψm for t m = 01, 04, and 10 wit = 005, t = 01 in Figure 62 Te figure sows tat α m > 0 and te computed solution ψm is clearly convex for all t m 62 Test 2 Te goal of tis test is to calculate te rate of convergence of ψ ε ψ ε and α ε α ε for a fixed ε wile varying t and wit te relation t = 2 We solve 313) 315) but wit a new boundary condition: ψε = φ ε, were φ ε is a known, given ν function Let Vg N and Wg D be defined in te same way as in Test 1 using te following

24 24 X FENG AND M NEILAN Fig 61 Test 1: Cange of ψ t m) ψ m wrt ε = 0023, t = 00005, tm = 025 test functions and parameters ct) = ψ ε, 1), g N = te tx2 1 +x2 2 )/2 x 1 ν x1 + x 2 ν x2 ), g D = t tx x 2 2))e tx2 1 +x2 2 ), εt 2 e tx2 1 +x2 2 )/ tx x 2 2) + t 2 x x 2 2) 2 ), F = t 2 + 4tx x 2 2) + t 2 x x 2 2) 2) e tx2 1 +x2 2 ) εt 2 etx2 1 +x2 2 )/ x x 2 2)t + 16t 2 x x 2 2) 2 + t 3 x x 2 2) 3), φ ε = 4x 1 t 2 + x 2 t 3 x x 2 2) ) ν x1 + 4x 2 t 2 + x 2 t 3 x x 2 2)ν x2 )) e tx 2 1 +x2 2 )/2, so tat te exact solution of 16) 112) is given by ψ ε x, t) = e tx2 1 +x2 2 )/2, α ε x, t) = t tx x 2 2))e tx2 1 +x2 2 ) εt 2 e tx2 1 +x2 2 )/ tx x 2 2) + t 2 x x 2 2) 2 ) Te errors at time t m = 025 are listed in Table 1 and are plotted verses t in Figure 63 Te results clearly indicate tat α ε t m ) α m L 2 = O t) and ψε t m ) ψ m = O t) in all norms as expected by te analysis in te previous section

25 A modified caracteristic finite element metod for te semigeostropic flows 25 Fig 62 Computed α m left) and ψm t = 01, = 005 rigt) for Test 1 at tm = 01 top), tm = 04 middle), and tm = 10 bottom) 63 Test 3 For tis test, we solve problem 313) 315) in te domain U = 0, 6) 2 and initial condition α 0 x) = 1 8 χ [2,4] [225,375]4 x 1 )x 1 2)375 x 2 )x 2 225),

26 26 X FENG AND M NEILAN Fig 63 Test 2: Cange of ψ ε t M ) ψ M wrt t = 2 ε = 001, t m = 025 t ψ ε t m) ψ m L 2 ψ ε t m) ψ m H 1 ψ ε t m) ψ m H 2 α ε t m) α m L E E E E E Table 1: Cange of ψ ε t M ) ψ M wrt t = 2 ε = 001, t m = 025 were χ [2,4] [225,375] denotes te caracteristic function of te set [2, 4] [225, 375] We comment tat te exact solution of tis problem is unknown We plot te computed α m and ψm at times t m = 0, t m = 005, and t m = 01 in Figure 64 wit parameters t = 0001, = 005, and ε = 001 Since we are not using piecewise linear functions to form W, we do not expect α m to be nonnegative for all m 0 cf Lemma 33) However, Figure 64 sows tat te deviation from zero is very small and tat ψ m is locally convex in te interior of U 7 Conclusions In tis paper, we ave developed a modified caracteristic finite element metod for a fully nonlinear formulation of B J Hoskins semigeostropic flow equations, wic models large scale flows wit frontogenesis Te system consists of a fully nonlinear Monge-Ampère equation and a transport equation Te main difficult for numerical approximations of te system is caused by te full nonlinearity of te Monge- Ampère equation To overcome tis difficult, we first introduce a vanising moment

27 A modified caracteristic finite element metod for te semigeostropic flows 27 Fig 64 Test 3: Computed α m left) and ψm t = 001, = 005, ε = 001 rigt) at tm = 0 top), tm = 005 middle), and tm = 01 bottom) approximation at te PDE level) of te semigeostropic flow equations, wic involves approximating te fully nonlinear second-order) Monge-Ampère equation by a family of fourt-order quasilinear equations We ten construct a modified caracteristic finite element metod for te regularized problem Te proposed metod consists of a conforming finite element discretization for te regularized Monge-Ampère equation and a modified