A P k+2 polynomial lifting operator on polygons and polyhedrons
AA P K +2 POLYNOMIAL LIFTING OPERATOR ON POLYGONS ANDPOLYHEDRONS
XIU YE ∗ AND
SHANGYOU ZHANG † Abstract. A P k +2 polynomial lifting operator is defined on polygons and polyhedrons. Itlifts discontinuous polynomials inside the polygon/polyhedron and on the faces to a one-piece P k +2 polynomial. With this lifting operator, we prove that the weak Galerkin finite element solution, afterthis lifting, converges at two orders higher than the optimal order, in both L and H norms. Thetheory is confirmed by numerical solutions of 2D and 3D Poisson equations. Key words. weak Galerkin, finite element methods, Poisson, polytopal meshes
AMS subject classifications.
Primary: 65N15, 65N30; Secondary: 35J50
1. Introduction.
In weak Galerkin finite element methods [12, 13], discontinu-ous polynomials, u defined inside each element and u b defined on each face of element,are employed to form an approximation space. In particular, on triangular/tetrahedralgrids, the P k - P k +1 ( P k inside a triangle, P k +1 on an edge) weak Galerkin finite ele-ment solution is two-order superconvergent in both L and H -like norms [2]. Further,with a careful construction of weak gradient, such P k - P k +1 weak Galerkin finite ele-ment is also two-order superconvergent on general polygonal and polyhedral meshes[14]. Here the super-convergence is defined for the difference between finite elementsolution u and the local L projection Q h u of the exact solution.In this paper, we construct a P k +2 polynomial lifting operator. It lifts an ( n + 1)-piece polynomial, { u , u b } , on a n -polygon/polyhedron T to a one-piece P k +2 poly-nomial on T . After such a lifting/post-processing, the weak Galerkin finite elementsolution is two-order super-convergent to the exact solution, i.e., (cid:107) u − u h (cid:107) + h | u − u h | ,h ≤ Ch k +1 | u | k +1 , (cid:107) u − L h u h (cid:107) + h | u − L h u h | ,h ≤ Ch k +3 | u | k +3 , where u h and u are the finite element solution and the exact solution, respectively,and h is the mesh size.This polynomial lifting operator is different from traditional polynomial liftingoperators [1, 3, 4, 5, 10]. These operators only lift a polynomial trace on the boundaryof an element to a polynomial inside the element, stably, i.e., subject to the minimumor a small energy. But here we lift both trace data and interior data to a polynomial,subject to the P k +2 accuracy. Additionally, even the trace (of boundary polynomials)is discontinuous here. Well, such a discontinuous-trace polynomial lifting is studiedin [7, 8, 9], but for H (curl) and H (div) polynomial lifting.
2. Weak Galerkin finite element.
For solving a model Poisson equation, − ∆ u = f in Ω , (2.1) u = 0 on ∂ Ω , (2.2) ∗ Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204([email protected]). This research was supported in part by National Science Foundation Grant DMS-1620016. † Department of Mathematical Sciences, University of Delaware, Newark, DE 19716([email protected]). 1 a r X i v : . [ m a t h . NA ] S e p here Ω is a polytopal domain in R or R , we subdivide the domain into shape-regular polygons/polyhedrons of size h , T h . For polynomial degree k ≥
1, we definethe weak Galerkin finite element spaces by(2.3) V h = { v h = { v , v b } : v | T ∈ P k ( T ) , v b | e ∈ P k +1 ( e ) , e ⊂ ∂T , T ∈ T h } and(2.4) V h = { v h : v h ∈ V h , v b = 0 on e ⊂ ∂ Ω } . The weak Galerkin finite element function assumes one d -dimensional P k polynomialinside each element T , and one ( d − P k +1 polynomial on each faceedge/polygon e .On an element T ∈ T h , we define the weak gradient ∇ w v h of a weak function v h = { v , v b } ∈ V h by the solution of polynomial equation on T :(2.5) (cid:90) T ∇ w v h q d x = (cid:90) ∂T v b q · n dS − (cid:90) T v ∇ · q d x ∀ q ∈ Λ k ( T ) , where Λ k ( T ) is a piece-wise polynomial space, but with one piece polynomial diver-gence and one piece polynomial trace on each face, on a sub-triangular/tetrahedralsubdivision of T = { T i , i = 1 , ..., n } ,Λ k ( T ) = { q ∈ H (div , T ) : q | T i ∈ P dk +1 ( T i ) , T i ⊂ T, ∇ · q ∈ P k ( T ) , q · n | e ∈ P k +1 ( e ) } . Here n is a fixed normal vector on edge/polygon e . To get a simplicial subdivision on T , some face edges/polygons have to be subdivided. That is, in addition to T = ∪ i T i , e = ∪ j e j , where { e j } is the set of face edges/triangles of { T i } .A weak Galerkin finite element approximation for (2.1)-(2.2) is defined by theunique solution u h = { u , u b } ∈ V h satisfying(2.6) ( ∇ w u h , ∇ w v h ) = ( f, v ) ∀ v h = { v , v b } ∈ V h . In [14], both (2.5) and (2.6) are proved to have a unique solution.
Theorem 2.1. ([14]) Let u and u h be the solutions of (2.1) and (2.6) , respec-tively. The following two-order superconvergence holds (cid:107) Q h u − u h (cid:107) + h ||| Q h u − u h ||| ≤ Ch k +3 | u | k +3 , (2.7) where Q h u = { Q u, Q b u } ∈ V h ( Q and Q b are local L -projection on T and e respectively), and ||| v h ||| = ( ∇ w v h , ∇ w v h ) / .
3. A P k +2 polynomial lifting operator. On an m -face polygon/polyhedron T we have ( m + 1) pieces of polynomials from a weak Galerkin finite element function.We need to lift these polynomials to a one-piece P k +2 polynomial, preserving P k +2 polynomials in the sense that L h Q h u = u if u is a P k +2 polynomial. Theorem 3.1.
The local L projection Q h : u ∈ P k +2 ( T ) → u h = { Q u, Q b u } ∈ V h is an injection, i.e., Q h u = 0 if and only if u = 0 . roof . Let u ∈ P k +2 ( T ) and Q h u = 0. For any vector polynomial q k +1 ∈ [ P k +1 ( T ] , we have (cid:90) T ∇ u · q k +1 d x = (cid:88) e ⊂ ∂T (cid:90) e u q k +1 · n dS − (cid:90) T u ∇ · q k +1 d x = (cid:88) e ⊂ ∂T (cid:90) e Q b u q k +1 · n dS − (cid:90) T Q u ∇ · q k +1 d x = 0 . Thus ∇ u = everywhere and u = C . Since Q u = 0, C = 0 and u = 0. Theorem 3.2.
The P k +2 polynomial lifting operator L h , defined in (3.3) below,is P k +2 polynomial preserving in the sense that L h Q h u = u, if u ∈ P k +2 ( T ) . (3.1) Consequently we have (cid:107) u − L h Q h u (cid:107) + h | u − L h Q h u | ,h ≤ Ch k +3 | u | k +3 . (3.2) Proof . Let P h : u h = { u , u b } ∈ V h → { ( P h u h ) , ( P h u h ) b } ∈ V h be the local,discrete L ( T ) projection on to the image space Q h P k +2 ( T ), i.e., (cid:90) T ( P h u h ) Q p k +2 d x + (cid:88) e ⊂ ∂T (cid:90) e ( P h u h ) b Q b p k +2 dS = (cid:90) T u Q p k +2 d x + (cid:88) e ⊂ ∂T (cid:90) e u b Q b p k +2 dS ∀ p k +2 ( T ) . The above equation has a unique solution as the left hand side bilinear form is coercive.By last theorem, Q h is one-to-one from P k +2 ( T ) on to the image space P h V h . Itsinverse defines an unique lifting operator: L h u h = Q − h ( P h u h ) ∈ (cid:89) T ∈T h P k +2 ( T ) . (3.3)By definition, (3.1) holds. Further, because L h Q h is a stable, local preserving P k +2 polynomial operator, by [11], it is an optimal-order interpolation operator and (3.2)holds. Theorem 3.3.
Let u and u h be the solutions of (2.1) and (2.6) , respectively.Then (cid:107) u − L h u h (cid:107) + h | u − L h u h | ,h ≤ Ch k +3 | u | k +3 , where | u | ,h = (cid:80) T ∈T h ( ∇ u, ∇ u ) .Proof . Noting the weak gradient of u h − P h u h is a piece-wise higher order, ||| · ||| -orthogonal polynomial over the polynomial ∇ L h u h , we have | L h u h | ,h = ||| P h u h ||| = ||| u h ||| − ||| ( I − P h ) u h ||| ≤ ||| u h ||| . y the triangle inequality, (3.2) and (2.7), | u − L h u h | ,h ≤ | u − L h Q h u | ,h + | L h ( Q h u − u h ) | ,h ≤ Ch k +3 | u | k +3 + ||| Q h u − u h |||≤ Ch k +2 | u | k +3 . By the finite dimensional norm equivalence with scaling, the trace inequality and thedefinition of weak gradient, we have (cid:107) L h u h (cid:107) ≤ C (cid:88) T ∈T h (cid:16) (cid:107) P h u (cid:107) T + 2 h (cid:107) P h ( u − u b ) (cid:107) ∂T (cid:17) ≤ Ch (cid:107) L h u h (cid:107) ||| ( I − P h ) u h ||| . By the triangle inequality, (3.2) and (2.7), we get (cid:107) u − L h u h (cid:107) ≤ (cid:107) u − L h Q h u (cid:107) + (cid:107) L h ( Q h u − u h ) (cid:107) ≤ Ch k +3 | u | k +3 . Fig. 4.1 . The first three levels of quadrilateral grids, for Table 4.1.
Table 4.1
Errors and orders of convergence by the P - P WG finite element on quadrilateral grids shownin Figure 4.1 for (4.1) . level (cid:107) u − u h (cid:107) rate (cid:107) Q h u − u h (cid:107) rate (cid:107) u − L h u h (cid:107) rate5 0.7356E-03 2.00 0.9360E-06 4.00 0.1308E-05 4.006 0.1838E-03 2.00 0.5851E-07 4.00 0.8178E-07 4.007 0.4595E-04 2.00 0.3663E-08 4.00 0.5116E-08 4.00 | u − u h | ,h rate ||| Q h u − u h ||| rate | u − L h u h | ,h rate5 0.5049E-01 1.00 0.2156E-03 3.00 0.2101E-03 3.006 0.2524E-01 1.00 0.2696E-04 3.00 0.2627E-04 3.007 0.1262E-01 1.00 0.3371E-05 3.00 0.3284E-05 3.00
4. Numerical Experiments.
We solve the 2D Poisson equation (2.1) on theunit square domain. The exact solution is chosen as u = sin( πx ) sin( πy ) . (4.1)We compute the solution (4.1) on a perturbed quadrilateral grids, shown in Figure4.1. We have two orders of superconvergence in L -norm and in H -like norm, shown able 4.2 Errors and orders of convergence by the P - P WG finite element on quadrilateral grids shownin Figure 4.1 for (4.1) . level (cid:107) u − u h (cid:107) rate (cid:107) Q h u − u h (cid:107) rate (cid:107) u − L h u h (cid:107) rate4 0.2229E-03 3.00 0.7659E-06 4.98 0.8555E-06 4.985 0.2787E-04 3.00 0.2404E-07 4.99 0.2682E-07 5.006 0.3484E-05 3.00 0.7521E-09 5.00 0.8390E-09 5.00 | u − u h | ,h rate ||| Q h u − u h ||| rate | u − L h u h | ,h rate4 0.1293E-01 2.00 0.1487E-03 3.99 0.9441E-04 3.995 0.3233E-02 2.00 0.9307E-05 4.00 0.5911E-05 4.006 0.8084E-03 2.00 0.5819E-06 4.00 0.3696E-06 4.00in Tables 4.1-4.2. In particular, the error after lifting is two orders higher than thatof the original error.Next we solve again the 2D Poisson equation (2.1) on the unit square domain withexact solution (4.1). We use quadrilateral-pentagon-hexagon hybrid grids, shown inFigure 4.2. Again the error after lifting is two orders higher, shown in Table 4.3. Fig. 4.2 . The first three levels of mixed-polygon grids, for Tables 4.3.
Table 4.3
Errors and orders of convergence, by the P - P WG finite element on mixed-polygon gridsshown in Figure 4.2 for (4.1) . level (cid:107) u − u h (cid:107) rate (cid:107) Q h u − u h (cid:107) rate (cid:107) u − L h u h (cid:107) rate5 0.8444E-03 2.00 0.1504E-05 4.00 0.1973E-05 4.006 0.2110E-03 2.00 0.9406E-07 4.00 0.1234E-06 4.007 0.5273E-04 2.00 0.5875E-08 4.00 0.7707E-08 4.00 | u − u h | ,h rate ||| Q h u − u h ||| rate | u − L h u h | ,h rate5 0.5891E-01 1.00 0.4798E-03 3.00 0.3272E-03 3.006 0.2945E-01 1.00 0.6001E-04 3.00 0.4090E-04 3.007 0.1472E-01 1.00 0.7502E-05 3.00 0.5113E-05 3.00Finally we solve the 3D Poisson equation (2.1) on the unit cube, with exactsolution u = sin( πx ) sin( πy ) sin( πz ) . (4.2)We use a wedge-type grids shown in Figure 4.3. The lifted finite element solution hastwo orders of superconvergence, shown in Table 4.4. (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) (cid:0)(cid:0) (cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64)(cid:64) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64)(cid:64)(cid:64) Fig. 4.3 . The first three levels of wedge grids used in Table 4.4.
Table 4.4
Errors and orders of convergence, by the P - P WG finite element on 3D wedge-type polyhedralgrids shown in Figure 4.3 for (4.2) . level (cid:107) u − u h (cid:107) rate (cid:107) Q h u − u h (cid:107) rate (cid:107) u − L h u h (cid:107) rate4 0.9655E-02 2.0 0.1608E-03 3.9 0.2626E-03 3.95 0.2398E-02 2.0 0.1022E-04 4.0 0.1658E-04 4.06 0.5987E-03 2.0 0.6419E-06 4.0 0.1039E-05 4.0 | u − u h | ,h rate ||| Q h u − u h ||| rate | u − L h u h | ,h rate4 0.2289E+00 1.0 0.2500E-01 3.0 0.1269E-01 3.05 0.1145E+00 1.0 0.3136E-02 3.0 0.1595E-02 3.06 0.5724E-01 1.0 0.3923E-03 3.0 0.1997E-03 3.0 REFERENCES[1] M. Ainsworth and C. Parker, H2