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Numerical Solution of Elliptic Equations with Finite Difference Methods

Learn about the 2D Finite Difference Method and Taylor's Theorem for numerical solutions of partial differential equations in science and engineering. Explore stability estimates, maximum principles, and error analysis in discrete and continuous problems.

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Numerical Solution of Elliptic Equations with Finite Difference Methods

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  1. -1 -1 4 -1 -1 5 point-scheme 5 point stencil 2D Finite Difference

  2. Taylor Expansions in 2d THEOREM—Taylor’s Theorem If u and its first n derivatives are continuous on the closed interval between a and a+h, and is differentiable on the open interval between a and a+h, then there exists a number c between a and a+hsuch that Taylor’s 2D

  3. 2D Finite Difference Lapidus, Leon, and George F. Pinder. Numerical solution of partial differential equations in science and engineering. John Wiley & Sons, 2011.

  4. Maximum Principle (Continuous Problem) (DMP) Discrete Maximum Principle (PT) Positive type scheme (DD) Diagonally Dominant the sum of the absolute values of the off-diagonal elements in one row is bounded by the diagonal element in that row Example

  5. Chapter 4: Finite Difference Methods for Elliptic Equations We consider Example Maximum Principle Consider the differential operator , and assume that and c = 0 its maximum is located at (1,1) Ifin Ω Remarks: 1) the maximum of u is attained at the boundary 2) if u has a maximum at an interior point of Ω, then u is constant 3) min principle is reduced to the max principle by looking at −u. Stability Estimate(w.r.t the max-norm) If there is a constant C such that The constant C depends on the coefficients of Abut not on u. Remarks: RHS of the above inequality contains only the data of the problem

  6. Chapter 4: Finite Difference Methods for Elliptic Equations We consider Example its maximum is located at (1,1) Discrete Maximum Principle If is such that attains its max for some Proof: assmume max at interior  max at all neibhor for Remark: 1) the maximum principle implies a stability estimate Stability Estimate For any mesh-function there is a constant C such that Proof: Remarks: The constant C independent of U and h. but not on u.

  7. Truncation Errors Remarks: Truncation Error This measures how well the exact solution of the differential equation (which we do not know) satisfies the difference equations

  8. Error Estimate define: error Theorem 4.2(Error Estimate) truncation Let be the solution of 2 2 1 1 Let be the solution of stability Then stability Independent of h as h 0 Proof: Apply stability result on Then use truncation error Theorem stability plus consistency implies convergence

  9. Maximum Principle (Continuous Problem) satisfies (DMP) Discrete Maximum Principle (PT) of positive type (DD) Diagonally Dominant the sum of the absolute values of the off-diagonal elements in one row is bounded by the diagonal element in that row Example

  10. curved boundary

  11. curved boundary

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