Spline Solution of Partial Differential Equations
We use bivariate and trivariate spline functions to solve 2D and 3D partial
differential equations. Some of advantages of spline solutions of PDE's are
1) piecewise polynomials of higher degrees can be used for numerical solution
of PDE's very easily; 2) spline solutions of any smooth can be used for
numerical solution very easily without constructing macro-elements; 3) Spline
solution may be variable smoothness across the domain; 4) Spline solution
can be obtained over arbitrary polygonal domains. We are able to solve many
different PDE's including 2D and 3D Navier-Stokes equations. Here we present
some standard example of PDE's.
1. Example of 2D Poisson Equations
Consider the following 2D Poisson equation with Dirichlet boundary condition:
-\Delta u= 40\exp(-(x^2+y^2))(1-x^2-y^2), (x,y) in [-2, 2]
x[-2, 2]
u(x,y)=\exp(x+y), (x,y) in \partial [-2, 2]x[-2, 2].
$$
The solution is relatively high active inside the domain
comparing with values near the boundary. Our spline solutions
can approximate it very well.
We use a triangulation with 25 vertices and 32 triangles.
We test many spline spaces. We list the
maximum errors of spline solutions against the exact solution.
The maximum errors are computed based on 101x 101 equally-spaced points
over [-2, 2]\times [-2, 2]. CPU times are computed using a PC with 128
MB memory and 400 mghz speed.
2. Example of 3D Poisson Equations
Consider a 3D Poisson equation with Dirichlet boundary condition
with exact solution u(x,y,z)=10\exp(-(x^2+y^2+z^2))
over an octahedron \Omega:=\langle (1,0,0),
(0,1,0),(-1,0,0), (0,-1,0),
$(0,0,1),(0,0,-1)\rangle$. We split \Omega into 8
tetrahedra by three coordinate planes. Let $\triangle$ denote the collection of
all 8 tetrahedra. We find approximate weak solution from
3D spline spaces $S^1_d(\triangle)$ for d=7, 8, 9, 10.
The maximum errors are computed based on 20x20x20
equally-spaced points over $\Omega$.
3. Example of 2D Biharmonic Equations
Consider a 2D biharmonic equation with exact
solution u(x,y)=\exp(x+y) over a unit square domain:
\Delta^2 u= 4\exp(x+y) & $(x,y)\in [0, 1]\times [0, 1]
u(x,y)=\exp(x+y), (x,y)\in \partial [0,1]\times [0, 1]
{\partial\over \partial x}u(x,y)=\exp(x+y),
(x,y)\in \partial [0,1]\times [0, 1]
{\partial\over \partial y}u(x,y)=\exp(x+y),
(x,y)\in \partial [0,1]\times [0, 1]$.
We use a triangulation with 25 vertices and 32 triangles. We list the
maximum errors of approximate spline solutions against the exact solution.
The maximum errors are computed based on $101\times 101$ equally-spaced points
over [0,1]x [0, 1].
Consider a 2D biharmonic equation with exact solution
$u(x,y)=10\exp(-(x^2+y^2))$ over a unit circular domain:
\Delta^2 u= 160\exp(-(x^2+y^2))(x^4+y^4+2x^2y^2+2-4x^2-4y^2),
(x,y)\in \{(x,y), x^2+y^2< 1\}
u(x,y)=10\exp(-(x^2+y^2)), (x,y)\in \{(x,y), x^2+y^2=1\}
{\partial \over \partial x}u(x,y)=-20x \exp(-(x^2+y^2)),
(x,y)\in \{(x,y), x^2+y^2=1\}
{\partial \over \partial y}u(x,y)=-20y \exp(-(x^2+y^2)),
(x,y)\in \{(x,y), x^2+y^2=1\}$.
We use the following triangulation and test many spline spaces.
The maximum errors of approximate spline solutions against the exact solution
are given in the following table.
The maximum errors are computed based on $101\times 101$ equally-spaced
points over $[-1, 1]\times [-1 1]$ within the circular domain.
