superconvergence

[qk11853]

from firedrake import *

nx = 4 mesh = UnitSquareMesh(nx, nx, diagonal="right") Uspace = FunctionSpace(mesh, "CG", 2) u = TrialFunction(Uspace) v = TestFunction(Uspace) n = FacetNormal(mesh) x, y = SpatialCoordinate(mesh)

bcs = DirichletBC(Uspace, cos(pi x) sin(pi * y), (1, 2, 3, 4))

u_ex = Function(Uspace).interpolate(cos(pi x) sin(pi y)) f = Function(Uspace).interpolate(2 pi * 2 cos(pi x) sin(pi * y))

a = inner(grad(u), grad(v)) dx l = f v * dx

uu = Function(Uspace)

solve(a == l, uu, bcs)

Why do I get superconvergent results when calculating poisson problems?

[colinjcotter]

Probably symmetric cancellations on the structured grid (perhaps combined with symmetric solution). Try breaking some of the symmetries.

[qk11853]

Probably symmetric cancellations on the structured grid (perhaps combined with symmetric solution). Try breaking some of the symmetries.

I'm not quite sure what you mean. How can I solve it? Thank you very much for your reply!

[colinjcotter]

There is nothing to solve? Error analysis provides an upper bound in the general case. Sometimes we are lucky.

[qk11853]

There is nothing to solve? Error analysis provides an upper bound in the general case. Sometimes we are lucky. For the finite element, P1 polynomial space, we get second-order convergence in the L2 sense and first-order convergence in the H1 sense. Now what I get with firedrake is second order and second order, how do I fix it?

[colinjcotter]

This is not inconsistent with the theory

[qk11853]

This is not inconsistent with the theory

But this superconvergent result, how can I get a result that's not superconvergent?

[colinjcotter]

Try a less symmetric case

On 9 Jun 2024, at 10:58, qk11853 @.***> wrote:



This is not inconsistent with the theory

But this superconvergent result, how can I get a result that's not superconvergent?

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